[genius] Sat Apr 29 16:58:25 2017 Jiri (George) Lebl <jirka 5z com>
- From: George Lebl <jirka src gnome org>
- To: commits-list gnome org
- Cc:
- Subject: [genius] Sat Apr 29 16:58:25 2017 Jiri (George) Lebl <jirka 5z com>
- Date: Sat, 29 Apr 2017 21:59:15 +0000 (UTC)
commit a453eb3e05b8b9c398b3e09c5aa3aa18e8e385b8
Author: Jiri (George) Lebl <jiri lebl gmail com>
Date: Sat Apr 29 16:58:27 2017 -0500
Sat Apr 29 16:58:25 2017 Jiri (George) Lebl <jirka 5z com>
* help/make-makefile-am.sh, help/update-xml-to-txt-html.sh, */*.html,
Makefile.am: Since automake is smarter than I am and knows how I
should work, I made a script to generate a Makefile.am so that it
can't complain. Now runs through distcheck with no errors!
ChangeLog | 7 +
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help/update-xml-to-txt-html.sh | 8 +
232 files changed, 8417 insertions(+), 877 deletions(-)
---
diff --git a/ChangeLog b/ChangeLog
index 1057e09..69587c9 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,3 +1,10 @@
+Sat Apr 29 16:58:25 2017 Jiri (George) Lebl <jirka 5z com>
+
+ * help/make-makefile-am.sh, help/update-xml-to-txt-html.sh, */*.html,
+ Makefile.am: Since automake is smarter than I am and knows how I
+ should work, I made a script to generate a Makefile.am so that it
+ can't complain. Now runs through distcheck with no errors!
+
Thu Apr 27 18:30:33 2017 Jiri (George) Lebl <jirka 5z com>
* configure.ac, src/Makefile.am: extra warning flags are handled
diff --git a/help/C/ch01.html b/help/C/ch01.html
new file mode 100644
index 0000000..a64287d
--- /dev/null
+++ b/help/C/ch01.html
@@ -0,0 +1,28 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 1.
Introduction</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="index.html" title="Genius Manual"><link
rel="prev" href="index.html" title="Genius Manual"><link rel="next" href="ch02.html" title="Chapter 2.
Getting Started"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Chapter 1. Introduction</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="index.html">Prev</a> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a
accesskey="n" href="ch02.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-introduction"></a>Chapte
r 1. Introduction</h1></div></div></div><p>
+ The <span class="application">Genius Mathematics Tool</span> application is a general calculator for
use as a desktop
+ calculator, an educational tool in mathematics, and is useful even for
+ research. The language used in <span class="application">Genius Mathematics Tool</span> is designed
to be
+ ‘mathematical’ in the sense that it should be ‘what
+ you mean is what you get’. Of course that is not an
+ entirely attainable goal. <span class="application">Genius Mathematics Tool</span> features
rationals, arbitrary
+ precision integers and multiple precision floats using the GMP library.
+ It handles complex numbers using cartesian notation. It has good
+ vector and matrix manipulation and can handle basic linear algebra.
+ The programming language allows user defined functions, variables and
+ modification of parameters.
+ </p><p>
+ <span class="application">Genius Mathematics Tool</span> comes in two versions. One version is the
graphical GNOME
+ version, which features an IDE style interface and the ability
+ to plot functions of one or two variables.
+ The command line version does not require GNOME, but of course
+ does not implement any feature that requires the graphical interface.
+ </p><p>
+ Parts of this manual describe the graphical version of the calculator,
+ but the language is of course the same. The command line only version
+ lacks the graphing capabilities and all other capabilities that require
+ the graphical user interface.
+ </p><p>
+ Generally, when some feature of the language (function, operator, etc...)
+ is new in some version past 1.0.5, it is mentioned, but
+ below 1.0.5 you would have to look at the NEWS file.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%"
align="left"><a accesskey="p" href="index.html">Prev</a> </td><td width="20%" align="center"> </td><td
width="40%" align="right"> <a accesskey="n" href="ch02.html">Next</a></td></tr><tr><td width="40%"
align="left" valign="top">Genius Manual </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Chapter 2. Getting
Started</td></tr></table></div></body></html>
diff --git a/help/C/ch02.html b/help/C/ch02.html
new file mode 100644
index 0000000..fd03db1
--- /dev/null
+++ b/help/C/ch02.html
@@ -0,0 +1,26 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 2. Getting
Started</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="index.html" title="Genius Manual"><link
rel="prev" href="ch01.html" title="Chapter 1. Introduction"><link rel="next" href="ch02s02.html" title="When
You Start Genius"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Chapter 2. Getting Started</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch01.html">Prev</a> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a
accesskey="n" href="ch02s02.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-getting-s
tarted"></a>Chapter 2. Getting Started</h1></div></div></div><div class="toc"><p><b>Table of
Contents</b></p><dl class="toc"><dt><span class="sect1"><a href="ch02.html#genius-to-start">To Start <span
class="application">Genius Mathematics Tool</span></a></span></dt><dt><span class="sect1"><a
href="ch02s02.html">When You Start Genius</a></span></dt></dl></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-to-start"></a>To Start
<span class="application">Genius Mathematics Tool</span></h2></div></div></div><p>You can start <span
class="application">Genius Mathematics Tool</span> in the following ways:
+ </p><div class="variablelist"><dl class="variablelist"><dt><span class="term"><span
class="guimenu">Applications</span> menu</span></dt><dd><p>
+ Depending on your operating system and version, the
+ menu item for <span class="application">Genius Mathematics Tool</span> could appear in a number
of different
+ places. It can be in the
+ <span class="guisubmenu">Education</span>,
+ <span class="guisubmenu">Accessories</span>,
+ <span class="guisubmenu">Office</span>,
+ <span class="guisubmenu">Science</span>, or
+ similar submenu, depending on your particular setup.
+ The menu item name you are looking for is
+ <span class="guimenuitem">Genius Math Tool</span>. Once you locate
+ this menu item click on it to start <span class="application">Genius Mathematics Tool</span>.
+ </p></dd><dt><span class="term"><span class="guilabel">Run</span> dialog</span></dt><dd><p>
+ Depending on your system installation the menu item
+ may not be available. If it is not, you can open the Run dialog
+ and execute <span class="command"><strong>gnome-genius</strong></span>.
+ </p></dd><dt><span class="term">Command line</span></dt><dd><p>
+ To start the GNOME version of <span class="application">Genius Mathematics Tool</span> execute
+ <span class="command"><strong>gnome-genius</strong></span> from the command line.
+ </p><p>
+ To start the command line only version,
+ execute the following command: <span class="command"><strong>genius</strong></span>.
+ This version does not include the graphical environment
+ and some functionality such as plotting will not be available.
+ </p></dd></dl></div></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch01.html">Prev</a>
</td><td width="20%" align="center"> </td><td width="40%" align="right"> <a accesskey="n"
href="ch02s02.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Chapter 1. Introduction
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> When You Start Genius</td></tr></table></div></body></html>
diff --git a/help/C/ch02s02.html b/help/C/ch02s02.html
new file mode 100644
index 0000000..7bce0dd
--- /dev/null
+++ b/help/C/ch02s02.html
@@ -0,0 +1,43 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>When You Start
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch02.html" title="Chapter 2. Getting
Started"><link rel="prev" href="ch02.html" title="Chapter 2. Getting Started"><link rel="next"
href="ch03.html" title="Chapter 3. Basic Usage"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">When You Start Genius</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch02.html">Prev</a> </td><th width="60%" align="center">Chapter 2. Getting
Started</th><td width="20%" align="right"> <a accesskey="n"
href="ch03.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" s
tyle="clear: both"><a name="genius-when-start"></a>When You Start Genius</h2></div></div></div><p>When you
start the GNOME edition of
+ <span class="application">Genius Mathematics Tool</span>, the window pictured in <a class="xref"
href="ch02s02.html#mainwindow-fig" title="Figure 2.1. Genius Mathematics Tool Window">Figure 2.1, “<span
class="application">Genius Mathematics Tool</span> Window”</a> is displayed.</p><div class="figure"><a
name="mainwindow-fig"></a><p class="title"><b>Figure 2.1. <span class="application">Genius Mathematics
Tool</span> Window</b></p><div class="figure-contents"><div class="screenshot"><div class="mediaobject"><img
src="figures/genius_window.png" alt="Shows Genius Mathematics Tool main window. Contains titlebar, menubar,
toolbar and working area. Menubar contains File, Edit, Calculator, Examples, Programs, Settings, and Help
menus."></div></div></div></div><br class="figure-break"><p>The <span class="application">Genius Mathematics
Tool</span> window contains the following elements:
+ </p><div class="variablelist"><dl class="variablelist"><dt><span
class="term">Menubar.</span></dt><dd><p>The menus on the menubar contain all of the commands that you need to
work with files in <span class="application">Genius Mathematics Tool</span>.
+ The <span class="guilabel">File</span> menu contains items for loading and saving items and
creating
+ new programs. The <span class="guilabel">Load and Run...</span> command does not open a new
window for
+ the program, but just executes the program directly. It is equivalent to the <span
class="command"><strong>load</strong></span>
+ command.</p><p>
+ The <span class="guilabel">Calculator</span> menu controls the
+calculator engine. It allows you to run the currently selected program or to
+interrupt the current calculation. You can also look at the full expression of
+the last answer (useful if the last answer was too large to fit onto the
+console), or you can view a listing of the values of all user defined
+variables. You can also monitor user variables, which is especially useful
+while a long calculation is running, or to debug a certain program.
+ Finally the <span class="guilabel">Calculator</span> allows plotting functions using a
user friendly dialog box.
+ </p><p>
+ The <span class="guilabel">Examples</span> menu is a list of example
+ programs or demos. If you open the menu, it will load the
+ example into a new program, which you can run, edit, modify,
+ and save. These programs should be well documented
+ and generally demonstrate either some feature of <span class="application">Genius
Mathematics Tool</span>
+ or some mathematical concept.
+ </p><p>
+ The <span class="guilabel">Programs</span> menu lists
+ the currently open programs and allows you to switch
+ between them.
+ </p><p>
+ The other menus have same familiar functions as in other applications.
+ </p></dd><dt><span class="term">Toolbar.</span></dt><dd><p>The toolbar contains a subset of the
commands that you can access from the menubar.</p></dd><dt><span class="term">Working area</span></dt><dd><p>
+ The working area is the primary method of interacting with
+ the application.
+ </p><p>
+ The working area initially has just the <span class="guilabel">Console</span> tab, which is
+ the main way of interacting with the calculator. Here you
+ type expressions and the results are immediately returned
+ after you hit the Enter key.
+ </p><p>
+ Alternatively you can write longer programs and those can
+ appear in separate tabs. The programs are a set of commands or
+ functions that can be run all at once rather than entering them
+ at the command line. The programs can be saved in files for later
+ retrieval.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch02.html">Prev</a> </td><td width="20%"
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diff --git a/help/C/ch03.html b/help/C/ch03.html
new file mode 100644
index 0000000..315fc10
--- /dev/null
+++ b/help/C/ch03.html
@@ -0,0 +1,44 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 3. Basic
Usage</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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rel="prev" href="ch02s02.html" title="When You Start Genius"><link rel="next" href="ch03s02.html" title="To
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align="center">Chapter 3. Basic Usage</th></tr><tr><td width="20%" align="left"><a accesskey="p"
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accesskey="n" href="ch03s02.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-usage"></a>Ch
apter 3. Basic Usage</h1></div></div></div><div class="toc"><p><b>Table of Contents</b></p><dl
class="toc"><dt><span class="sect1"><a href="ch03.html#genius-usage-workarea">Using the Work
Area</a></span></dt><dt><span class="sect1"><a href="ch03s02.html">To Create a New Program
</a></span></dt><dt><span class="sect1"><a href="ch03s03.html">To Open and Run a Program
</a></span></dt></dl></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-usage-workarea"></a>Using the Work Area</h2></div></div></div><p>
+ Normally you interact with the calculator in the <span class="guilabel">Console</span> tab of the
+ work area. If you are running the text only version then the console
+ will be the only thing that is available to you. If you want to use
+ <span class="application">Genius Mathematics Tool</span> as a calculator only, just type in your
expression in the console, it
+ will be evaluated, and the returned value will be printed.
+ </p><p>
+ To evaluate an expression, type it into the <span class="guilabel">Console</span> work area and
press enter.
+ Expressions are written in a
+language called GEL. The most simple GEL expressions just looks like
+mathematics. For example
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>30*70 +
67^3.0 + ln(7) * (88.8/100)</code></strong>
+</pre><p>
+or
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>62734 +
812634 + 77^4 mod 5</code></strong>
+</pre><p>
+or
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>| sin(37) -
e^7 |</code></strong>
+</pre><p>
+or
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>sum n=1 to 70
do 1/n</code></strong>
+</pre><p>
+(Last is the harmonic sum from 1 to 70)
+</p><p>
+To get a list of functions and commands, type:
+</p><pre class="screen"><code class="prompt">genius> </code><strong
class="userinput"><code>help</code></strong>
+</pre><p>
+If you wish to get more help on a specific function, type:
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>help
FunctionName</code></strong>
+</pre><p>
+To view this manual, type:
+</p><pre class="screen"><code class="prompt">genius> </code><strong
class="userinput"><code>manual</code></strong>
+</pre><p>
+</p><p>
+Suppose you have previously saved some GEL commands as a program to a file and
+you now want to execute them.
+To load this program from the file <code class="filename">path/to/program.gel</code>,
+type
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>load
path/to/program.gel</code></strong>
+</pre><p>
+<span class="application">Genius Mathematics Tool</span> keeps track of the current directory.
+To list files in the current directory type <span class="command"><strong>ls</strong></span>, to change
directory
+do <strong class="userinput"><code>cd directory</code></strong> as in the UNIX command shell.
+</p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch02s02.html">Prev</a> </td><td width="20%" align="center">
</td><td width="40%" align="right"> <a accesskey="n" href="ch03s02.html">Next</a></td></tr><tr><td
width="40%" align="left" valign="top">When You Start Genius </td><td width="20%" align="center"><a
accesskey="h" href="index.html">Home</a></td><td width="40%" align="right" valign="top"> To Create a New
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diff --git a/help/C/ch03s02.html b/help/C/ch03s02.html
new file mode 100644
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+++ b/help/C/ch03s02.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>To Create a New
Program</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
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href="ch03.html">Prev</a> </td><th width="60%" align="center">Chapter 3. Basic Usage</th><td width="20%"
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class="sect1"><div class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a name="genius-usage-create-program"></a>To Create a New Program
</h2></div></div></div><p>
+ If you wish to enter several more complicated commands, or perhaps write a complicated
+ function using the <a class="link" href="ch05.html" title="Chapter 5. GEL Basics">GEL</a>
language, you can create a new
+ program.
+ </p><p>
+To start writing a new program, choose
+<span class="guimenu">File</span> → <span class="guimenuitem">New
+Program</span>. A new tab will appear in the work area. You
+can write a <a class="link" href="ch05.html" title="Chapter 5. GEL Basics">GEL</a> program in this work area.
+Once you have written your program you can run it by
+<span class="guimenu">Calculator</span> → <span class="guimenuitem">Run</span> (or
+the <span class="guilabel">Run</span> toolbar button).
+This will execute your program and will display any output on the <span class="guilabel">Console</span> tab.
+Executing a program is equivalent of taking the text of the program and
+typing it into the console. The only difference is that this input is done
+independent of the console and just the output goes onto the console.
+<span class="guimenu">Calculator</span> → <span class="guimenuitem">Run</span>
+will always run the currently selected program even if you are on the <span class="guilabel">Console</span>
+tab. The currently selected program has its tab in bold type. To select a
+program, just click on its tab.
+ </p><p>
+To save the program you've just written, choose <span class="guimenu">File</span> → <span
class="guimenuitem">Save As...</span>.
+Similarly as in other programs you can choose
+<span class="guimenu">File</span> → <span class="guimenuitem">Save</span> to save a program that already has
+a filename attached to it. If you have many opened programs you have edited and wish to save you can also
choose
+<span class="guimenu">File</span> → <span class="guimenuitem">Save All Unsaved</span>.
+ </p><p>
+ Programs that have unsaved changes will have a "[+]" next to their filename. This way you can
see if the file
+ on disk and the currently opened tab differ in content. Programs which have not yet had a
filename associated
+ with them are always considered unsaved and no "[+]" is printed.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch03.html">Prev</a> </td><td width="20%" align="center"><a
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</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
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diff --git a/help/C/ch03s03.html b/help/C/ch03s03.html
new file mode 100644
index 0000000..5a0a95d
--- /dev/null
+++ b/help/C/ch03s03.html
@@ -0,0 +1,16 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>To Open and Run a
Program</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch03.html" title="Chapter 3. Basic Usage"><link
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title="Chapter 4. Plotting"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">To Open and Run a Program </th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch03s02.html">Prev</a> </td><th width="60%" align="center">Chapter 3. Basic Usage</th><td width="20%"
align="right"> <a accesskey="n" href="ch04.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-usage-open-program"></a>To Open and Run a Program
</h2></div></div></div><p>
+To open a file, choose
+<span class="guimenu">File</span> → <span class="guimenuitem">Open</span>.
+A new tab containing the file will appear in the work area. You can use this to
+edit the file.
+ </p><p>
+To run a program from a file, choose
+<span class="guimenu">File</span> → <span class="guimenuitem">Load and
+Run...</span>. This will run the program without opening it
+in a separate tab. This is equivalent to the <span class="command"><strong>load</strong></span> command.
+ </p><p>
+ If you have made edits to a file you wish to throw away and want to reload to the version
that's on disk,
+ you can choose the
+ <span class="guimenu">File</span> → <span class="guimenuitem">Reload from Disk</span> menuitem.
This is useful for experimenting
+ with a program and making temporary edits, to run a program, but that you do not intend to keep.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch03s02.html">Prev</a> </td><td width="20%" align="center"><a
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href="ch04.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">To Create a New Program
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
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diff --git a/help/C/ch04.html b/help/C/ch04.html
new file mode 100644
index 0000000..a0a96db
--- /dev/null
+++ b/help/C/ch04.html
@@ -0,0 +1,47 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 4.
Plotting</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="index.html" title="Genius Manual"><link
rel="prev" href="ch03s03.html" title="To Open and Run a Program"><link rel="next" href="ch04s02.html"
title="Parametric Plots"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Chapter 4. Plotting</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch03s03.html">Prev</a> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a
accesskey="n" href="ch04s02.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-gel-plotting"></a>Chap
ter 4. Plotting</h1></div></div></div><div class="toc"><p><b>Table of Contents</b></p><dl
class="toc"><dt><span class="sect1"><a href="ch04.html#genius-line-plots">Line Plots</a></span></dt><dt><span
class="sect1"><a href="ch04s02.html">Parametric Plots</a></span></dt><dt><span class="sect1"><a
href="ch04s03.html">Slopefield Plots</a></span></dt><dt><span class="sect1"><a
href="ch04s04.html">Vectorfield Plots</a></span></dt><dt><span class="sect1"><a href="ch04s05.html">Surface
Plots</a></span></dt></dl></div><p>
+ Plotting support is only available in the graphical GNOME version.
+ All plotting accessible from the graphical interface is available from
+ the <span class="guilabel">Create Plot</span> window. You can access this window by either clicking
+ on the <span class="guilabel">Plot</span> button on the toolbar or selecting <span
class="guilabel">Plot</span> from the <span class="guilabel">Calculator</span>
+ menu. You can also access the plotting functionality by using the
+ <a class="link" href="ch11s20.html" title="Plotting">plotting
+ functions</a> of the GEL language. See
+ <a class="xref" href="ch05.html" title="Chapter 5. GEL Basics">Chapter 5, <i>GEL Basics</i></a> to
find out how to
+ enter expressions that Genius understands.
+ </p><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-line-plots"></a>Line Plots</h2></div></div></div><p>
+ To graph real valued functions of one variable open the <span class="guilabel">Create Plot</span>
+ window. You can also use the
+ <a class="link" href="ch11s20.html#gel-function-LinePlot"><code class="function">LinePlot</code></a>
function
+ on the command line (see its documentation).
+ </p><p>
+ Once you click the <span class="guilabel">Plot</span> button, a window opens up with some notebooks
in it.
+ You want to be in the <span class="guilabel">Function line plot</span> notebook
+tab, and inside you want to be on the <span class="guilabel">Functions / Expressions</span> notebook tab.
+See <a class="xref" href="ch04.html#lineplot-fig" title="Figure 4.1. Create Plot Window">Figure 4.1, “Create
Plot Window”</a>.
+ </p><div class="figure"><a name="lineplot-fig"></a><p class="title"><b>Figure 4.1. Create Plot
Window</b></p><div class="figure-contents"><div class="screenshot"><div class="mediaobject"><img
src="figures/line_plot.png" alt="Shows the line plotting window."></div></div></div></div><br
class="figure-break"><p>
+ Type expressions with <strong class="userinput"><code>x</code></strong> as
+ the independent variable into the textboxes. Alternatively you can give names of functions such as
+ <strong class="userinput"><code>cos</code></strong> rather then having to type <strong
class="userinput"><code>cos(x)</code></strong>.
+ You can graph up to ten functions. If you make a mistake and Genius cannot
+ parse the input it will signify this with a warning icon on the right of the text
+ input box where the error occurred, as well as giving you an error dialog.
+ You can change the ranges of the dependent and independent variables in the bottom
+ part of the dialog.
+ The <code class="varname">y</code> (dependent) range can be set automatically by turning on the <span
class="guilabel">Fit dependent axis</span>
+ checkbox.
+ The names of the variables can also be changed.
+ Pressing the <span class="guilabel">Plot</span> button produces the graph shown in <a class="xref"
href="ch04.html#lineplot2-fig" title="Figure 4.2. Plot Window">Figure 4.2, “Plot Window”</a>.
+ </p><p>
+ The variables can be renamed by clicking the <span class="guilabel">Change variable
names...</span> button, which is useful if you wish to print or save the figure and don't want to use the
standard
+ names. Finally you can also avoid printing the legend and the axis labels completely,
+ which is also useful if printing or
+ saving, when the legend might simply be clutter.
+ </p><div class="figure"><a name="lineplot2-fig"></a><p class="title"><b>Figure 4.2. Plot
Window</b></p><div class="figure-contents"><div class="screenshot"><div class="mediaobject"><img
src="figures/line_plot_graph.png" alt="The graph produced."></div></div></div></div><br
class="figure-break"><p>
+ From here you can print out the plot, create encapsulated postscript
+ or a PNG version of the plot or change the zoom. If the dependent axis was
+ not set correctly you can have Genius fit it by finding out the extrema of
+ the graphed functions.
+ </p><p>
+ For plotting using the command line see the documentation of the
+ <a class="link" href="ch11s20.html#gel-function-LinePlot"><code class="function">LinePlot</code></a>
function.
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch03s03.html">Prev</a> </td><td width="20%" align="center">
</td><td width="40%" align="right"> <a accesskey="n" href="ch04s02.html">Next</a></td></tr><tr><td
width="40%" align="left" valign="top">To Open and Run a Program </td><td width="20%" align="center"><a
accesskey="h" href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Parametric
Plots</td></tr></table></div></body></html>
diff --git a/help/C/ch04s02.html b/help/C/ch04s02.html
new file mode 100644
index 0000000..e45fbcb
--- /dev/null
+++ b/help/C/ch04s02.html
@@ -0,0 +1,21 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Parametric
Plots</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch04.html" title="Chapter 4. Plotting"><link
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href="ch04.html">Prev</a> </td><th width="60%" align="center">Chapter 4. Plotting</th><td width="20%"
align="right"> <a accesskey="n" href="ch04s03.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="geniu
s-parametric-plots"></a>Parametric Plots</h2></div></div></div><p>
+ In the create plot window, you can also choose the <span class="guilabel">Parametric</span> notebook
+ tab to create two dimensional parametric plots. This way you can
+ plot a single parametric function. You can either specify the
+ points as <code class="varname">x</code> and <code class="varname">y</code>, or giving a single
complex number
+ as a function of the variable <code class="varname">t</code>.
+ The range of the variable <code class="varname">t</code> is given explicitly, and the function is
sampled
+ according to the given increment.
+ The <code class="varname">x</code> and <code class="varname">y</code> range can be set
+ automatically by turning on the <span class="guilabel">Fit dependent axis</span>
+ checkbox, or it can be specified explicitly.
+ See <a class="xref" href="ch04s02.html#paramplot-fig" title="Figure 4.3. Parametric Plot Tab">Figure
4.3, “Parametric Plot Tab”</a>.
+ </p><div class="figure"><a name="paramplot-fig"></a><p class="title"><b>Figure 4.3. Parametric Plot
Tab</b></p><div class="figure-contents"><div class="screenshot"><div class="mediaobject"><img
src="figures/parametric.png" alt="Parametric plotting tab in the Create Plot
window."></div></div></div></div><br class="figure-break"><p>
+ An example of a parametric plot is given in
+ <a class="xref" href="ch04s02.html#paramplot2-fig" title="Figure 4.4. Parametric Plot">Figure 4.4,
“Parametric Plot”</a>.
+ Similar operations can be
+ done on such graphs as can be done on the other line plots.
+ For plotting using the command line see the documentation of the
+ <a class="link" href="ch11s20.html#gel-function-LinePlotParametric"><code
class="function">LinePlotParametric</code></a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotCParametric"><code
class="function">LinePlotCParametric</code></a> function.
+ </p><div class="figure"><a name="paramplot2-fig"></a><p class="title"><b>Figure 4.4. Parametric
Plot</b></p><div class="figure-contents"><div class="screenshot"><div class="mediaobject"><img
src="figures/parametric_graph.png" alt="Parametric plot produced"></div></div></div></div><br
class="figure-break"></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch04.html">Prev</a> </td><td width="20%" align="center"><a
accesskey="u" href="ch04.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch04s03.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Chapter 4. Plotting
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Slopefield Plots</td></tr></table></div></body></html>
diff --git a/help/C/ch04s03.html b/help/C/ch04s03.html
new file mode 100644
index 0000000..7060fab
--- /dev/null
+++ b/help/C/ch04s03.html
@@ -0,0 +1,23 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Slopefield
Plots</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch04.html" title="Chapter 4. Plotting"><link
rel="prev" href="ch04s02.html" title="Parametric Plots"><link rel="next" href="ch04s04.html"
title="Vectorfield Plots"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Slopefield Plots</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch04s02.html">Prev</a> </td><th width="60%" align="center">Chapter 4. Plotting</th><td width="20%"
align="right"> <a accesskey="n" href="ch04s04.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="gen
ius-slopefield-plots"></a>Slopefield Plots</h2></div></div></div><p>
+ In the create plot window, you can also choose the <span class="guilabel">Slope field</span> notebook
+ tab to create a two dimensional slope field plot.
+ Similar operations can be
+ done on such graphs as can be done on the other line plots.
+ For plotting using the command line see the documentation of the
+ <a class="link" href="ch11s20.html#gel-function-SlopefieldPlot"><code
class="function">SlopefieldPlot</code></a> function.
+ </p><p>
+ When a slope field is active, there is an extra <span class="guilabel">Solver</span> menu available,
+ through which you can bring up the solver dialog. Here you can have Genius plot specific
+ solutions for the given initial conditions. You can either specify initial conditions in the dialog,
+ or you can click on the plot directly to specify the initial point. While the solver dialog
+ is active, the zooming by clicking and dragging does not work. You have to close the dialog first
+ if you want to zoom using the mouse.
+ </p><p>
+ The solver uses the standard Runge-Kutta method.
+ The plots will stay on the screen until cleared. The solver will stop whenever it reaches the
boundary
+ of the plot window. Zooming does not change the limits or parameters of the solutions,
+ you will have to clear and redraw them with appropriate parameters.
+ You can also use the
+ <a class="link" href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>
+ function to draw solutions from the command line or programs.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch04s02.html">Prev</a> </td><td width="20%" align="center"><a
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width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
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diff --git a/help/C/ch04s04.html b/help/C/ch04s04.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Vectorfield
Plots</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch04.html" title="Chapter 4. Plotting"><link
rel="prev" href="ch04s03.html" title="Slopefield Plots"><link rel="next" href="ch04s05.html" title="Surface
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align="right"> <a accesskey="n" href="ch04s05.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="geniu
s-vectorfield-plots"></a>Vectorfield Plots</h2></div></div></div><p>
+ In the create plot window, you can also choose the <span class="guilabel">Vector field</span> notebook
+ tab to create a two dimensional vector field plot.
+ Similar operations can be
+ done on such graphs as can be done on the other line plots.
+ For plotting using the command line see the documentation of the
+ <a class="link" href="ch11s20.html#gel-function-VectorfieldPlot"><code
class="function">VectorfieldPlot</code></a> function.
+ </p><p>
+ By default the direction and magnitude of the vector field is shown.
+ To only show direction and not the magnitude, check the appropriate
+ checkbox to normalize the arrow lengths.
+ </p><p>
+ When a vector field is active, there is an extra <span class="guilabel">Solver</span> menu available,
+ through which you can bring up the solver dialog. Here you can have Genius plot specific
+ solutions for the given initial conditions. You can either specify initial conditions in the dialog,
+ or you can click on the plot directly to specify the initial point. While the solver dialog
+ is active, the zooming by clicking and dragging does not work. You have to close the dialog first
+ if you want to zoom using the mouse.
+ </p><p>
+ The solver uses the standard Runge-Kutta method.
+ The plots will stay on the screen until cleared.
+ Zooming does not change the limits or parameters of the solutions,
+ you will have to clear and redraw them with appropriate parameters.
+ You can also use the
+ <a class="link" href="ch11s20.html#gel-function-VectorfieldDrawSolution"><code
class="function">VectorfieldDrawSolution</code></a>
+ function to draw solutions from the command line or programs.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch04s03.html">Prev</a> </td><td width="20%" align="center"><a
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width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
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diff --git a/help/C/ch04s05.html b/help/C/ch04s05.html
new file mode 100644
index 0000000..d934298
--- /dev/null
+++ b/help/C/ch04s05.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Surface
Plots</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch04.html" title="Chapter 4. Plotting"><link
rel="prev" href="ch04s04.html" title="Vectorfield Plots"><link rel="next" href="ch05.html" title="Chapter 5.
GEL Basics"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Surface
Plots</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch04s04.html">Prev</a> </td><th
width="60%" align="center">Chapter 4. Plotting</th><td width="20%" align="right"> <a accesskey="n"
href="ch05.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-s
urface-plots"></a>Surface Plots</h2></div></div></div><p>
+ Genius can also plot surfaces. Select the <span class="guilabel">Surface plot</span> tab in the
+ main notebook of the <span class="guilabel">Create Plot</span> window. Here you can specify a single
+ expression that should use either <code class="varname">x</code> and <code class="varname">y</code>
as real independent variables
+ or <code class="varname">z</code> as a complex variable (where <code class="varname">x</code> is the
real part of <code class="varname">z</code> and <code class="varname">y</code> is the
+ imaginary part). For example to plot the modulus of the cosine
+ function for complex parameters,
+ you could enter <strong class="userinput"><code>|cos(z)|</code></strong>. This would be
+ equivalent to <strong class="userinput"><code>|cos(x+1i*y)|</code></strong>.
+ See <a class="xref" href="ch04s05.html#surfaceplot-fig" title="Figure 4.5. Surface Plot">Figure 4.5,
“Surface Plot”</a>.
+ For plotting using the command line see the documentation of the
+ <a class="link" href="ch11s20.html#gel-function-SurfacePlot"><code
class="function">SurfacePlot</code></a> function.
+ </p><p>
+ The <code class="varname">z</code> range can be set automatically by turning on the <span
class="guilabel">Fit dependent axis</span>
+ checkbox. The variables can be renamed by clicking the <span class="guilabel">Change variable
names...</span> button, which is useful if you wish to print or save the figure and don't want to use the
standard
+ names. Finally you can also avoid printing the legend, which is also useful if printing or
+ saving, when the legend might simply be clutter.
+ </p><div class="figure"><a name="surfaceplot-fig"></a><p class="title"><b>Figure 4.5. Surface
Plot</b></p><div class="figure-contents"><div class="screenshot"><div class="mediaobject"><img
src="figures/surface_graph.png" alt="Modulus of the complex cosine function."></div></div></div></div><br
class="figure-break"><p>
+ In surface mode, left and right arrow keys on your keyboard will rotate the
+ view along the z axis. Alternatively you can rotate along any axis by
+ selecting <span class="guilabel">Rotate axis...</span> in the <span
class="guilabel">View</span>
+ menu. The <span class="guilabel">View</span> menu also has a top view mode which rotates the
+ graph so that the z axis is facing straight out, that is, we view the graph from the top
+ and get essentially just the colors that define the values of the function getting a
+ temperature plot of the function. Finally you should
+ try <span class="guilabel">Start rotate animation</span>, to start a continuous slow rotation.
+ This is especially good if using <span class="application">Genius Mathematics Tool</span> to
present to an audience.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch04s04.html">Prev</a> </td><td width="20%" align="center"><a
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href="ch05.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Vectorfield Plots </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Chapter 5. GEL Basics</td></tr></table></div></body></html>
diff --git a/help/C/ch05.html b/help/C/ch05.html
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+++ b/help/C/ch05.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 5. GEL
Basics</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Chapter
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</td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch05s02.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-gel"></a>Chapter 5. GEL Basic
s</h1></div></div></div><div class="toc"><p><b>Table of Contents</b></p><dl class="toc"><dt><span
class="sect1"><a href="ch05.html#genius-gel-values">Values</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-numbers">Numbers</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-booleans">Booleans</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-strings">Strings</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-null">Null</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s02.html">Using Variables</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-setting">Setting Variables</a></span></dt><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-built-in">Built-in Variables</a></span></dt><dt><span
class="sect2"><a href="ch05s02.html#genius-gel-previous-result">Previous Result
Variable</a></span></dt></dl></dd><
dt><span class="sect1"><a href="ch05s03.html">Using Functions</a></span></dt><dd><dl><dt><span
class="sect2"><a href="ch05s03.html#genius-gel-functions-defining">Defining
Functions</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-variable-argument-lists">Variable Argument
Lists</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-passing-functions">Passing Functions to
Functions</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-operations">Operations on
Functions</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s04.html">Separator</a></span></dt><dt><span class="sect1"><a
href="ch05s05.html">Comments</a></span></dt><dt><span class="sect1"><a href="ch05s06.html">Modular
Evaluation</a></span></dt><dt><span class="sect1"><a href="ch05s07.html">List of GEL
Operators</a></span></dt></dl></div><p>
+ GEL stands for Genius Extension Language. It is the language you use
+ to write programs in Genius. A program in GEL is simply an
+ expression that evaluates to a number, a matrix, or another object
+ in GEL.
+ <span class="application">Genius Mathematics Tool</span> can be used as a simple calculator, or as a
+ powerful theoretical research tool. The syntax is meant to
+ have as shallow of a learning curve as possible, especially for use
+ as a calculator.
+ </p><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-values"></a>Values</h2></div></div></div><p>
+ Values in GEL can be <a class="link" href="ch05.html#genius-gel-values-numbers"
title="Numbers">numbers</a>, <a class="link" href="ch05.html#genius-gel-values-booleans"
title="Booleans">Booleans</a>, or <a class="link" href="ch05.html#genius-gel-values-strings"
title="Strings">strings</a>. GEL also treats
+<a class="link" href="ch08.html" title="Chapter 8. Matrices in GEL">matrices</a> as values.
+ Values can be used in calculations, assigned to variables and returned from functions, among
other uses.
+ </p><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-values-numbers"></a>Numbers</h3></div></div></div><p>
+Integers are the first type of number in GEL. Integers are written in the normal way.
+</p><pre class="programlisting">1234
+</pre><p>
+Hexadecimal and octal numbers can be written using C notation. For example:
+</p><pre class="programlisting">0x123ABC
+01234
+</pre><p>
+Or you can type numbers in an arbitrary base using <code class="literal"><base>\<number></code>.
Digits higher than 10 use letters in a similar way to hexadecimal. For example, a number in base 23 could be
written:
+</p><pre class="programlisting">23\1234ABCD
+</pre><p>
+ </p><p>
+The second type of GEL number is rationals. Rationals are simply achieved by dividing two integers. So one
could write:
+</p><pre class="programlisting">3/4
+</pre><p>
+to get three quarters. Rationals also accept mixed fraction notation. So in order to get one and three
tenths you could write:
+</p><pre class="programlisting">1 3/10
+</pre><p>
+ </p><p>
+The next type of number is floating point. These are entered in a similar fashion to C notation. You can use
<code class="literal">E</code>, <code class="literal">e</code> or <code class="literal">@</code> as the
exponent delimiter. Note that using the exponent delimiter gives a float even if there is no decimal point in
the number. Examples:
+</p><pre class="programlisting">1.315
+7.887e77
+7.887e-77
+.3
+0.3
+77e5
+</pre><p>
+ When Genius prints a floating point number it will always append a
+ <code class="computeroutput">.0</code> even if the number is whole. This is to indicate that
+ floating point numbers are taken as imprecise quantities. When a number is written in the
+ scientific notation, it is always a floating point number and thus Genius does not
+ print the <code class="computeroutput">.0</code>.
+ </p><p>
+The final type of number in GEL is the complex numbers. You can enter a complex number as a sum of real and
imaginary parts. To add an imaginary part, append an <code class="literal">i</code>. Here are examples of
entering complex numbers:
+</p><pre class="programlisting">1+2i
+8.01i
+77*e^(1.3i)
+</pre><p>
+ </p><div class="important" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Important</h3><p>
+When entering imaginary numbers, a number must be in front of the <code class="literal">i</code>. If you use
<code class="literal">i</code> by itself, Genius will interpret this as referring to the variable <code
class="varname">i</code>. If you need to refer to <code class="literal">i</code> by itself, use <code
class="literal">1i</code> instead.
+ </p><p>
+In order to use mixed fraction notation with imaginary numbers you must have the mixed fraction in
parentheses. (i.e., <strong class="userinput"><code>(1 2/5)i</code></strong>)
+ </p></div></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-values-booleans"></a>Booleans</h3></div></div></div><p>
+Genius also supports native Boolean values. The two Boolean constants are
+defined as <code class="constant">true</code> and <code class="constant">false</code>; these
+identifiers can be used like any other variable. You can also use the
+identifiers <code class="constant">True</code>, <code class="constant">TRUE</code>,
+<code class="constant">False</code> and <code class="constant">FALSE</code> as aliases for the
+above.
+ </p><p>
+At any place where a Boolean expression is expected, you can use a Boolean
+value or any expression that produces either a number or a Boolean. If
+Genius needs to evaluate a number as a Boolean it will interpret
+0 as <code class="constant">false</code> and any other number as
+<code class="constant">true</code>.
+ </p><p>
+In addition, you can do arithmetic with Boolean values. For example:
+</p><pre class="programlisting">( (1 + true) - false ) * true
+</pre><p>
+is the same as:
+</p><pre class="programlisting">( (true or true) or not false ) and true
+</pre><p>
+Only addition, subtraction and multiplication are supported. If you mix numbers with Booleans in an
expression then the numbers are converted to Booleans as described above. This means that, for example:
+</p><pre class="programlisting">1 == true
+</pre><p>
+always evaluates to <code class="constant">true</code> since 1 will be converted to <code
class="constant">true</code> before being compared to <code class="constant">true</code>.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-values-strings"></a>Strings</h3></div></div></div><p>
+Like numbers and Booleans, strings in GEL can be stored as values inside variables and passed to functions.
You can also concatenate a string with another value using the plus operator. For example:
+</p><pre class="programlisting">a=2+3;"The result is: "+a
+</pre><p>
+will create the string:
+</p><pre class="programlisting">The result is: 5
+</pre><p>
+You can also use C-like escape sequences such as <code class="literal">\n</code>,<code
class="literal">\t</code>,<code class="literal">\b</code>,<code class="literal">\a</code> and <code
class="literal">\r</code>. To get a <code class="literal">\</code> or <code class="literal">"</code> into the
string you can quote it with a <code class="literal">\</code>. For example:
+</p><pre class="programlisting">"Slash: \\ Quotes: \" Tabs: \t1\t2\t3"
+</pre><p>
+will make a string:
+</p><pre class="programlisting">Slash: \ Quotes: " Tabs: 1 2 3
+</pre><p>
+Do note however that when a string is returned from a function, escapes are
+quoted, so that the output can be used as input. If you wish to print the
+string as it is (without escapes), use the
+<a class="link" href="ch11s02.html#gel-function-print"><code class="function">print</code></a>
+or
+<a class="link" href="ch11s02.html#gel-function-printn"><code class="function">printn</code></a> functions.
+ </p><p>
+ In addition, you can use the library function <a class="link"
href="ch11s02.html#gel-function-string"><code class="function">string</code></a> to convert anything to a
string. For example:
+</p><pre class="programlisting">string(22)
+</pre><p>
+will return
+</p><pre class="programlisting">"22"
+</pre><p>
+Strings can also be compared with <code class="literal">==</code> (equal), <code class="literal">!=</code>
(not equal) and <code class="literal"><=></code> (comparison) operators
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-values-null"></a>Null</h3></div></div></div><p>
+There is a special value called
+<code class="constant">null</code>. No operations can be performed on
+it, and nothing is printed when it is returned. Therefore,
+<code class="constant">null</code> is useful when you do not want output from an
+expression. The value <code class="constant">null</code> can be obtained as an expression when you
+type <code class="literal">.</code>, the constant <code class="constant">null</code> or nothing.
+By nothing we mean that if you end an expression with
+a separator <code class="literal">;</code>, it is equivalent to ending it with a
+separator followed by a <code class="constant">null</code>.
+ </p><p>
+Example:
+</p><pre class="programlisting">x=5;.
+x=5;
+</pre><p>
+ </p><p>
+Some functions return <code class="constant">null</code> when no value can be returned
+or an error happened. Also <code class="constant">null</code> is used as an empty vector
+or matrix, or an empty reference.
+</p></div></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch04s05.html">Prev</a> </td><td width="20%" align="center">
</td><td width="40%" align="right"> <a accesskey="n" href="ch05s02.html">Next</a></td></tr><tr><td
width="40%" align="left" valign="top">Surface Plots </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Using
Variables</td></tr></table></div></body></html>
diff --git a/help/C/ch05s02.html b/help/C/ch05s02.html
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+++ b/help/C/ch05s02.html
@@ -0,0 +1,45 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Using
Variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch05.html" title="Chapter 5. GEL Basics"><link
rel="prev" href="ch05.html" title="Chapter 5. GEL Basics"><link rel="next" href="ch05s03.html" title="Using
Functions"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Using
Variables</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch05.html">Prev</a> </td><th
width="60%" align="center">Chapter 5. GEL Basics</th><td width="20%" align="right"> <a accesskey="n"
href="ch05s03.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="ge
nius-gel-variables"></a>Using Variables</h2></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">VariableName
+</pre><p>
+Example:
+</p><pre class="screen"><code class="prompt">genius> </code><strong
class="userinput"><code>e</code></strong>
+= 2.71828182846
+</pre><p>
+ </p><p>
+To evaluate a variable by itself, just enter the name of the variable. This will return the value of the
variable. You can use a variable anywhere you would normally use a number or string. In addition, variables
are necessary when defining functions that take arguments (see <a class="xref"
href="ch05s03.html#genius-gel-functions-defining" title="Defining Functions">the section called “Defining
Functions”</a>).
+ </p><div class="tip" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Using Tab
completion</h3><p>
+You can use Tab completion to get Genius to complete variable names for you. Try typing the first few
letters of the name and pressing <strong class="userinput"><code>Tab</code></strong>.
+ </p></div><div class="important" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Variable names are case sensitive</h3><p>
+The names of variables are case sensitive. That means that variables named <code
class="varname">hello</code>, <code class="varname">HELLO</code> and <code class="varname">Hello</code> are
all different variables.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-variables-setting"></a>Setting Variables</h3></div></div></div><p>
+Syntax:
+</p><pre class="programlisting"><identifier> = <value>
+<identifier> := <value>
+</pre><p>
+Example:
+</p><pre class="programlisting">x = 3
+x := 3
+</pre><p>
+ </p><p>
+To assign a value to a variable, use the <code class="literal">=</code> or <code class="literal">:=</code>
operators. These operators set the value of the variable and return the value you set, so you can do things
like
+</p><pre class="programlisting">a = b = 5
+</pre><p>
+This will set <code class="varname">b</code> to 5 and then also set <code class="varname">a</code> to 5.
+ </p><p>
+The <code class="literal">=</code> and <code class="literal">:=</code> operators can both be used to set
variables. The difference between them is that the <code class="literal">:=</code> operator always acts as an
assignment operator, whereas the <code class="literal">=</code> operator may be interpreted as testing for
equality when used in a context where a Boolean expression is expected.
+ </p><p>
+ For more information about the scope of variables, that is when are what variables visible, see <a
class="xref" href="ch06s05.html" title="Global Variables and Scope of Variables">the section called “Global
Variables and Scope of Variables”</a>.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-variables-built-in"></a>Built-in Variables</h3></div></div></div><p>
+GEL has a number of built-in ‘variables’, such as
+<code class="varname">e</code>, <code class="varname">pi</code> or <code class="varname">GoldenRatio</code>.
These are widely used constants with a preset value, and
+they cannot be assigned new values.
+There are a number of other built-in variables.
+See <a class="xref" href="ch11s04.html" title="Constants">the section called “Constants”</a> for a full
list. Note that <code class="varname">i</code> is not by default
+the square root of negative one (the imaginary number), and is undefined to allow its use as a counter. If
you wish to write the imaginary number you need to
+use <strong class="userinput"><code>1i</code></strong>.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-previous-result"></a>Previous Result Variable</h3></div></div></div><p>
+The <code class="varname">Ans</code> and <code class="varname">ans</code> variables can be used to get the
result of the last expression. For example, if you had performed some calculation, to add 389 to the result
you could do:
+</p><pre class="programlisting">Ans+389
+</pre><p>
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch05.html">Prev</a> </td><td width="20%" align="center"><a
accesskey="u" href="ch05.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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diff --git a/help/C/ch05s03.html b/help/C/ch05s03.html
new file mode 100644
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--- /dev/null
+++ b/help/C/ch05s03.html
@@ -0,0 +1,74 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Using
Functions</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch05.html" title="Chapter 5. GEL Basics"><link
rel="prev" href="ch05s02.html" title="Using Variables"><link rel="next" href="ch05s04.html"
title="Separator"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Using Functions</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch05s02.html">Prev</a> </td><th width="60%" align="center">Chapter 5. GEL Basics</th><td width="20%"
align="right"> <a accesskey="n" href="ch05s04.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-gel
-functions"></a>Using Functions</h2></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">FunctionName(argument1, argument2, ...)
+</pre><p>
+Example:
+</p><pre class="programlisting">Factorial(5)
+cos(2*pi)
+gcd(921,317)
+</pre><p>
+
+To evaluate a function, enter the name of the function, followed by the arguments (if any) to the function
in parentheses. This will return the result of applying the function to its arguments. The number of
arguments to the function is, of course, different for each function.
+ </p><p>
+ There are many built-in functions, such as <a class="link"
href="ch11s06.html#gel-function-sin"><code class="function">sin</code></a>, <a class="link"
href="ch11s06.html#gel-function-cos"><code class="function">cos</code></a> and <a class="link"
href="ch11s06.html#gel-function-tan"><code class="function">tan</code></a>. You can use the <a class="link"
href="ch11.html#gel-command-help"><code class="function">help</code></a> built-in command to get a list of
available functions, or see <a class="xref" href="ch11.html" title="Chapter 11. List of GEL
functions">Chapter 11, <i>List of GEL functions</i></a> for a full listing.
+ </p><div class="tip" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Using Tab
completion</h3><p>
+You can use Tab completion to get Genius to complete function names for you. Try typing the first few
letters of the name and pressing <strong class="userinput"><code>Tab</code></strong>.
+ </p></div><div class="important" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Function names are case sensitive</h3><p>
+The names of functions are case sensitive. That means that functions named <code
class="function">dosomething</code>, <code class="function">DOSOMETHING</code> and <code
class="function">DoSomething</code> are all different functions.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-functions-defining"></a>Defining Functions</h3></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">function <identifier>(<comma separated arguments>) =
<function body>
+<identifier> = (`() = <function body>)
+</pre><p>
+The <code class="literal">`</code> is the backquote character, and signifies an anonymous function. By
setting it to a variable name you effectively define a function.
+ </p><p>
+A function takes zero or more comma separated arguments, and returns the result of the function body.
Defining your own functions is primarily a matter of convenience; one possible use is to have sets of
functions defined in GEL files that Genius can load in order to make them available.
+Example:
+</p><pre class="programlisting">function addup(a,b,c) = a+b+c
+</pre><p>
+then <strong class="userinput"><code>addup(1,4,9)</code></strong> yields 14
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-functions-variable-argument-lists"></a>Variable Argument Lists</h3></div></div></div><p>
+If you include <code class="literal">...</code> after the last argument name in the function declaration,
then Genius will allow any number of arguments to be passed in place of that argument. If no arguments were
passed then that argument will be set to <code class="constant">null</code>. Otherwise, it will be a
horizontal vector containing all the arguments. For example:
+</p><pre class="programlisting">function f(a,b...) = b
+</pre><p>
+Then <strong class="userinput"><code>f(1,2,3)</code></strong> yields <code
class="computeroutput">[2,3]</code>, while <strong class="userinput"><code>f(1)</code></strong> yields a
<code class="constant">null</code>.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-functions-passing-functions"></a>Passing Functions to Functions</h3></div></div></div><p>
+In Genius, it is possible to pass a function as an argument to another function. This can be done using
either ‘function nodes’ or anonymous functions.
+ </p><p>
+If you do not enter the parentheses after a function name, instead of being evaluated, the function will
instead be returned as a ‘function node’. The function node can then be passed to another function.
+Example:
+</p><pre class="programlisting">function f(a,b) = a(b)+1;
+function b(x) = x*x;
+f(b,2)
+</pre><p>
+ </p><p>
+To pass functions that are not defined,
+you can use an anonymous function (see <a class="xref" href="ch05s03.html#genius-gel-functions-defining"
title="Defining Functions">the section called “Defining Functions”</a>). That is, you want to pass a
function without giving it a name.
+Syntax:
+</p><pre class="programlisting">function(<comma separated arguments>) = <function body>
+`(<comma separated arguments>) = <function body>
+</pre><p>
+Example:
+</p><pre class="programlisting">function f(a,b) = a(b)+1;
+f(`(x) = x*x,2)
+</pre><p>
+This will return 5.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-functions-operations"></a>Operations on Functions</h3></div></div></div><p>
+ Some functions allow arithmetic operations, and some single argument functions such as <a
class="link" href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a> or <a class="link"
href="ch11s05.html#gel-function-ln"><code class="function">ln</code></a>, to operate on the function. For
example,
+</p><pre class="programlisting">exp(sin*cos+4)
+</pre><p>
+will return a function that takes <code class="varname">x</code> and returns <strong
class="userinput"><code>exp(sin(x)*cos(x)+4)</code></strong>. It is functionally equivalent
+to typing
+</p><pre class="programlisting">`(x) = exp(sin(x)*cos(x)+4)
+</pre><p>
+
+This operation can be useful when quickly defining functions. For example to create a function called <code
class="varname">f</code>
+to perform the above operation, you can just type:
+</p><pre class="programlisting">f = exp(sin*cos+4)
+</pre><p>
+It can also be used in plotting. For example, to plot sin squared you can enter:
+</p><pre class="programlisting">LinePlot(sin^2)
+</pre><p>
+ </p><div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Warning</h3><p>
+Not all functions can be used in this way. For example, when you use a binary operation the functions must
take the same number of arguments.
+ </p></div></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch05s02.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch05.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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diff --git a/help/C/ch05s04.html b/help/C/ch05s04.html
new file mode 100644
index 0000000..379468f
--- /dev/null
+++ b/help/C/ch05s04.html
@@ -0,0 +1,28 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Separator</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch05.html" title="Chapter 5. GEL Basics"><link rel="prev" href="ch05s03.html"
title="Using Functions"><link rel="next" href="ch05s05.html" title="Comments"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
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Basics</th><td width="20%" align="right"> <a accesskey="n"
href="ch05s05.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-gel-separator"><
/a>Separator</h2></div></div></div><p>
+ GEL is somewhat different from other languages in how it deals with multiple commands and
functions.
+ In GEL you must chain commands together with a separator operator.
+That is, if you want to type more than one expression you have to use
+the <code class="literal">;</code> operator in between the expressions. This is
+a way in which both expressions are evaluated and the result of the second one (or the last one
+if there is more than two expressions) is returned.
+Suppose you type the following:
+</p><pre class="programlisting">3 ; 5
+</pre><p>
+This expression will yield 5.
+ </p><p>
+This will require some parenthesizing to make it unambiguous sometimes,
+especially if the <code class="literal">;</code> is not the top most primitive. This slightly differs from
+other programming languages where the <code class="literal">;</code> is a terminator of statements, whereas
+in GEL it’s actually a binary operator. If you are familiar with pascal
+this should be second nature. However genius can let you pretend it is a
+terminator to some degree. If a <code class="literal">;</code> is found at the end of a parenthesis or a
block,
+genius will append a null to it as if you would have written
+<strong class="userinput"><code>;null</code></strong>.
+This is useful in case you do not want to return a value from say a loop,
+or if you handle the return differently. Note that it will slightly slow down
+the code if it is executed too often as there is one more operator involved.
+ </p><p>
+ If you are typing expressions in a program you do not have to add a semicolon. In this case
+ genius will simply print the return value whenever it executes the expression. However, do
note that if you are defining a
+ function, the body of the function is a single expression.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch05s03.html">Prev</a> </td><td width="20%" align="center"><a
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diff --git a/help/C/ch05s05.html b/help/C/ch05s05.html
new file mode 100644
index 0000000..b2e74b8
--- /dev/null
+++ b/help/C/ch05s05.html
@@ -0,0 +1,10 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Comments</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch05.html" title="Chapter 5. GEL Basics"><link rel="prev" href="ch05s04.html"
title="Separator"><link rel="next" href="ch05s06.html" title="Modular Evaluation"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Comments</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch05s04.html">Prev</a> </td><th width="60%"
align="center">Chapter 5. GEL Basics</th><td width="20%" align="right"> <a accesskey="n"
href="ch05s06.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-gel-comments">
</a>Comments</h2></div></div></div><p>
+ GEL is similar to other scripting languages in that <code class="literal">#</code> denotes
+ a comment, that is text that is not meant to be evaluated. Everything beyond the
+ pound sign till the end of line will just be ignored. For example,
+</p><pre class="programlisting"># This is just a comment
+# every line in a comment must have its own pound sign
+# in the next line we set x to the value 123
+x=123;
+</pre><p>
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch05s04.html">Prev</a> </td><td width="20%" align="center"><a
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Modular
Evaluation</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch05.html" title="Chapter 5. GEL Basics"><link
rel="prev" href="ch05s05.html" title="Comments"><link rel="next" href="ch05s07.html" title="List of GEL
Operators"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Modular
Evaluation</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch05s05.html">Prev</a> </td><th
width="60%" align="center">Chapter 5. GEL Basics</th><td width="20%" align="right"> <a accesskey="n"
href="ch05s07.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name=
"genius-gel-modular-evaluation"></a>Modular Evaluation</h2></div></div></div><p>
+ Genius implements modular arithmetic.
+To use it you just add "mod <integer>" after
+the expression. Example:
+<strong class="userinput"><code>2^(5!) * 3^(6!) mod 5</code></strong>
+It could be possible to do modular arithmetic by computing with integers and then modding in the end with
+the <code class="literal">%</code> operator, which simply gives the remainder, but
+that may be time consuming if not impossible when working with larger numbers.
+For example, <strong class="userinput"><code>10^(10^10) % 6</code></strong> will simply not work (the
exponent
+will be too large), while
+<strong class="userinput"><code>10^(10^10) mod 6</code></strong> is instantaneous. The first expression
first tries to compute the integer
+<strong class="userinput"><code>10^(10^10)</code></strong> and then find remainder after division by 6,
while the second expression evaluates
+everything modulo 6 to begin with.
+ </p><p>
+You can calculate the inverses of numbers mod some integer by just using
+rational numbers (of course the inverse has to exist).
+Examples:
+</p><pre class="programlisting">10^-1 mod 101
+1/10 mod 101</pre><p>
+You can also do modular evaluation with matrices including taking inverses,
+powers and dividing.
+Example:
+</p><pre class="programlisting">A = [1,2;3,4]
+B = A^-1 mod 5
+A*B mod 5</pre><p>
+This should yield the identity matrix as B will be the inverse of A mod 5.
+ </p><p>
+Some functions such as
+<a class="link" href="ch11s05.html#gel-function-sqrt"><code class="function">sqrt</code></a> or
+<a class="link" href="ch11s05.html#gel-function-log"><code class="function">log</code></a>
+work in a different way when in modulo mode. These will then work like their
+discrete versions working within the ring of integers you selected. For
+example:
+</p><pre class="programlisting">genius> sqrt(4) mod 7
+=
+[2, 5]
+genius> 2*2 mod 7
+= 4</pre><p>
+ <code class="function">sqrt</code> will actually return all the possible square
+ roots.
+ </p><p>
+ Do not chain mod operators, simply place it at the end of the computation, all computations in
the expression on the left
+ will be carried out in mod arithmetic. If you place a mod inside
+ a mod, you will get unexpected results. If you simply want to
+ mod a single number and control exactly when remainders are
+ taken, best to use the <code class="literal">%</code> operator. When you
+ need to chain several expressions in modular arithmetic with
+ different divisors, it may be best to just split up the expression into several and use
+ temporary variables to avoid a mod inside a mod.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch05s05.html">Prev</a> </td><td width="20%" align="center"><a
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diff --git a/help/C/ch05s07.html b/help/C/ch05s07.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>List of GEL
Operators</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch05.html" title="Chapter 5. GEL Basics"><link
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Programming with GEL"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">List of GEL Operators</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch05s06.html">Prev</a> </td><th width="60%" align="center">Chapter 5. GEL Basics</th><td width="20%"
align="right"> <a accesskey="n" href="ch06.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style=
"clear: both"><a name="genius-gel-operator-list"></a>List of GEL Operators</h2></div></div></div><p>
+ Everything in GEL is really just an expression. Expressions are stringed together with
+ different operators. As we have seen, even the separator is simply a binary operator
+ in GEL. Here is a list of the operators in GEL.
+ </p><div class="variablelist"><dl class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>
+ The separator, just evaluates both
+ <code class="varname">a</code> and
+ <code class="varname">b</code>,
+ but returns only the result of
+ <code class="varname">b</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>
+ The assignment operator. This assigns <code class="varname">b</code> to
+<code class="varname">a</code> (<code class="varname">a</code> must be a valid <a class="link"
href="ch06s09.html" title="Lvalues">lvalue</a>) (note however that this operator
+may be translated to <code class="literal">==</code> if used in a place where boolean
+expression is expected)
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a:=b</code></strong></span></dt><dd><p>
+ The assignment operator. Assigns <code class="varname">b</code> to
+<code class="varname">a</code> (<code class="varname">a</code> must be a valid <a class="link"
href="ch06s09.html" title="Lvalues">lvalue</a>). This is
+different from <code class="literal">=</code> because it never gets translated to a
+<code class="literal">==</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>|a|</code></strong></span></dt><dd><p>
+ Absolute value.
+ In case the expression is a complex number the result will be the modulus
+(distance from the origin). For example:
+<strong class="userinput"><code>|3 * e^(1i*pi)|</code></strong>
+returns 3.
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>
+ Exponentiation, raises <code class="varname">a</code> to the <code class="varname">b</code>th
power.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.^b</code></strong></span></dt><dd><p>
+ Element by element exponentiation. Raise each element of a matrix
+ <code class="varname">a</code> to the <code class="varname">b</code>th power. Or if
+ <code class="varname">b</code> is a matrix of the same size as
+ <code class="varname">a</code>, then do the operation element by element.
+ If <code class="varname">a</code> is a number and <code class="varname">b</code> is a
+ matrix then it creates matrix of the same size as
+ <code class="varname">b</code> with <code class="varname">a</code> raised to all the
+ different powers in <code class="varname">b</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a+b</code></strong></span></dt><dd><p>
+ Addition. Adds two numbers, matrices, functions or strings. If
+ you add a string to anything the result will just be a string. If one is
+ a square matrix and the other a number, then the number is multiplied by
+ the identity matrix.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a-b</code></strong></span></dt><dd><p>
+ Subtraction. Subtract two numbers, matrices or functions.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>
+ Multiplication. This is the normal matrix multiplication.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>
+ Element by element multiplication if <code class="varname">a</code> and
+ <code class="varname">b</code> are matrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a/b</code></strong></span></dt><dd><p>
+ Division. When <code class="varname">a</code> and <code class="varname">b</code> are just
numbers
+ this is the normal division. When they are matrices, then this is
+ equivalent to <strong class="userinput"><code>a*b^-1</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>
+ Element by element division. Same as <strong class="userinput"><code>a/b</code></strong>
for
+ numbers, but operates element by element on matrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>
+ Back division. That is this is the same as <strong class="userinput"><code>b/a</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>
+ Element by element back division.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>
+ The mod operator. This does not turn on the <a class="link" href="ch05s06.html" title="Modular
Evaluation">modular mode</a>, but
+ just returns the remainder of <strong class="userinput"><code>a/b</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
+ Element by element the mod operator. Returns the remainder
+ after element by element integer <strong class="userinput"><code>a./b</code></strong>.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>
+ Modular evaluation operator. The expression <code class="varname">a</code>
+ is evaluated modulo <code class="varname">b</code>. See <a class="xref" href="ch05s06.html"
title="Modular Evaluation">the section called “Modular Evaluation”</a>.
+ Some functions and operators behave differently modulo an integer.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>
+ Factorial operator. This is like
+ <strong class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>
+ Double factorial operator. This is like
+ <strong class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p>
+ Equality operator.
+ Returns <code class="constant">true</code> or <code class="constant">false</code>
+ depending on <code class="varname">a</code> and <code class="varname">b</code> being equal or
not.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a!=b</code></strong></span></dt><dd><p>
+ Inequality operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> does not
+ equal <code class="varname">b</code> else returns <code class="constant">false</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<>b</code></strong></span></dt><dd><p>
+ Alternative inequality operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> does not
+ equal <code class="varname">b</code> else returns <code class="constant">false</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></strong></span></dt><dd><p>
+ Less than or equal operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ less than or equal to
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a <= b <= c</code></strong> (can
+ also be combined with the less than operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>
+ Greater than or equal operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than or equal to
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
+ (can also be combine with the greater than operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
+ Less than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ less than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
+ (can also be combine with the less than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
+ Greater than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
+ (can also be combine with the greater than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>
+ Comparison operator. If <code class="varname">a</code> is equal to
+ <code class="varname">b</code> it returns 0, if <code class="varname">a</code> is less
+ than <code class="varname">b</code> it returns -1 and if
+ <code class="varname">a</code> is greater than <code class="varname">b</code> it
+ returns 1.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a and
b</code></strong></span></dt><dd><p>
+ Logical and. Returns true if both
+ <code class="varname">a</code> and <code class="varname">b</code> are true,
+ else returns false. If given numbers, nonzero numbers
+ are treated as true.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a or
b</code></strong></span></dt><dd><p>
+ Logical or.
+ Returns true if either
+ <code class="varname">a</code> or <code class="varname">b</code> is true,
+ else returns false. If given numbers, nonzero numbers
+ are treated as true.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
+ Logical xor.
+ Returns true exactly one of
+ <code class="varname">a</code> or <code class="varname">b</code> is true,
+ else returns false. If given numbers, nonzero numbers
+ are treated as true.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
+ Logical not. Returns the logical negation of <code class="varname">a</code>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
+ Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>
+ Variable referencing (to pass a reference to a variable).
+ See <a class="xref" href="ch06s08.html" title="References">the section called “References”</a>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>
+ Variable dereferencing (to access a referenced variable).
+ See <a class="xref" href="ch06s08.html" title="References">the section called “References”</a>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>
+ Matrix conjugate transpose. That is, rows and columns get swapped and we take complex
conjugate of all entries. That is
+ if the i,j element of <code class="varname">a</code> is x+iy, then the j,i element of
<strong class="userinput"><code>a'</code></strong> is x-iy.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>
+ Matrix transpose, does not conjugate the entries. That is,
+ the i,j element of <code class="varname">a</code> becomes the j,i element of <strong
class="userinput"><code>a.'</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>
+ Get element of a matrix in row <code class="varname">b</code> and column
+ <code class="varname">c</code>. If <code class="varname">b</code>,
+ <code class="varname">c</code> are vectors, then this gets the corresponding
+ rows columns or submatrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>
+ Get row of a matrix (or multiple rows if <code class="varname">b</code> is a vector).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>
+ Same as above.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>
+ Get column of a matrix (or columns if <code class="varname">c</code> is a
+ vector).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>
+ Same as above.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>
+ Get an element from a matrix treating it as a vector. This will
+ traverse the matrix row-wise.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>
+ Build a vector from <code class="varname">a</code> to <code class="varname">b</code> (or
specify a row, column region for the <code class="literal">@</code> operator). For example to get rows 2 to
4 of matrix <code class="varname">A</code> we could do
+ </p><pre class="programlisting">A@(2:4,)
+ </pre><p>
+ as <strong class="userinput"><code>2:4</code></strong> will return a vector
+ <strong class="userinput"><code>[2,3,4]</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a:b:c</code></strong></span></dt><dd><p>
+ Build a vector from <code class="varname">a</code> to <code class="varname">c</code>
+ with <code class="varname">b</code> as a step. That is for example
+ </p><pre class="programlisting">genius> 1:2:9
+=
+`[1, 3, 5, 7, 9]
+</pre><p>
+ </p><p>
+ When the numbers involved are floating point numbers, for example
+ <strong class="userinput"><code>1.0:0.4:3.0</code></strong>, the output is what is expected
+ even though adding 0.4 to 1.0 five times is actually just slightly
+ more than 3.0 due to the way that floating point numbers are
+ stored in base 2 (there is no 0.4, the actual number stored is
+ just ever so slightly bigger). The way this is handled is the
+ same as in the for, sum, and prod loops. If the end is within
+ <strong class="userinput"><code>2^-20</code></strong> times the step size of the endpoint,
+ the endpoint is used and we assume there were roundoff errors.
+ This is not perfect, but it handles the majority of the cases.
+ This check is done only from version 1.0.18 onwards, so execution
+ of your code may differ on older versions. If you want to avoid
+ dealing with this issue, use actual rational numbers, possibly
+ using the <code class="function">float</code> if you wish to get floating
+ point numbers in the end. For example
+ <strong class="userinput"><code>1:2/5:3</code></strong> does the right thing and
+ <strong class="userinput"><code>float(1:2/5:3)</code></strong> even gives you floating
+ point numbers and is ever so slightly more precise than
+ <strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
+ Make a imaginary number (multiply <code class="varname">a</code> by the
+ imaginary). Note that normally the number <code class="varname">i</code> is
+ written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
+ </p><pre class="programlisting">(a)*1i
+ </pre><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>
+ Quote an identifier so that it doesn't get evaluated. Or
+ quote a matrix so that it doesn't get expanded.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a swapwith
b</code></strong></span></dt><dd><p>
+ Swap value of <code class="varname">a</code> with the value
+ of <code class="varname">b</code>. Currently does not operate
+ on ranges of matrix elements.
+ It returns <code class="constant">null</code>.
+ Available from version 1.0.13.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>increment
a</code></strong></span></dt><dd><p>
+ Increment the variable <code class="varname">a</code> by 1. If
+ <code class="varname">a</code> is a matrix, then increment each element.
+ This is equivalent to <strong class="userinput"><code>a=a+1</code></strong>, but
+ it is somewhat faster. It returns <code class="constant">null</code>.
+ Available from version 1.0.13.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>increment a by
b</code></strong></span></dt><dd><p>
+ Increment the variable <code class="varname">a</code> by <code class="varname">b</code>.
If
+ <code class="varname">a</code> is a matrix, then increment each element.
+ This is equivalent to <strong class="userinput"><code>a=a+b</code></strong>, but
+ it is somewhat faster. It returns <code class="constant">null</code>.
+ Available from version 1.0.13.
+ </p></dd></dl></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Note</h3><p>
+The @() operator makes the : operator most useful. With this you can specify regions of a matrix. So that
a@(2:4,6) is the rows 2,3,4 of the column 6. Or a@(,1:2) will get you the first two columns of a matrix. You
can also assign to the @() operator, as long as the right value is a matrix that matches the region in size,
or if it is any other type of value.
+</p></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Note</h3><p>
+The comparison operators (except for the <=> operator, which behaves normally), are not strictly
binary operators, they can in fact be grouped in the normal mathematical way, e.g.: (1<x<=y<5) is a
legal boolean expression and means just what it should, that is (1<x and x≤y and y<5)
+</p></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Note</h3><p>
+The unitary minus operates in a different fashion depending on where it
+appears. If it appears before a number it binds very closely, if it appears in
+front of an expression it binds less than the power and factorial operators.
+So for example <strong class="userinput"><code>-1^k</code></strong> is really <strong
class="userinput"><code>(-1)^k</code></strong>,
+but <strong class="userinput"><code>-foo(1)^k</code></strong> is really <strong
class="userinput"><code>-(foo(1)^k)</code></strong>. So
+be careful how you use it and if in doubt, add parentheses.
+</p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch05s06.html">Prev</a> </td><td width="20%" align="center"><a
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 6. Programming
with GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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class="titlepage"><div><div><h1 class="title"><a name="genius-gel-programmi
ng"></a>Chapter 6. Programming with GEL</h1></div></div></div><div class="toc"><p><b>Table of
Contents</b></p><dl class="toc"><dt><span class="sect1"><a
href="ch06.html#genius-gel-conditionals">Conditionals</a></span></dt><dt><span class="sect1"><a
href="ch06s02.html">Loops</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-while">While Loops</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-for">For Loops</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-foreach">Foreach Loops</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-break-continue">Break and Continue</a></span></dt></dl></dd><dt><span
class="sect1"><a href="ch06s03.html">Sums and Products</a></span></dt><dt><span class="sect1"><a
href="ch06s04.html">Comparison Operators</a></span></dt><dt><span class="sect1"><a href="ch06s05.html">Global
Variables and Scope of Variables</a></span></dt><dt><span
class="sect1"><a href="ch06s06.html">Parameter variables</a></span></dt><dt><span class="sect1"><a
href="ch06s07.html">Returning</a></span></dt><dt><span class="sect1"><a
href="ch06s08.html">References</a></span></dt><dt><span class="sect1"><a
href="ch06s09.html">Lvalues</a></span></dt></dl></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-conditionals"></a>Conditionals</h2></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">if <expression1> then <expression2> [else <expression3>]
+</pre><p>
+If <code class="literal">else</code> is omitted, then if the <code class="literal">expression1</code> yields
<code class="constant">false</code> or 0, <code class="literal">NULL</code> is returned.
+ </p><p>
+Examples:
+</p><pre class="programlisting">if(a==5)then(a=a-1)
+if b<a then b=a
+if c>0 then c=c-1 else c=0
+a = ( if b>0 then b else 1 )
+</pre><p>
+Note that <code class="literal">=</code> will be translated to <code class="literal">==</code> if used
inside the expression for <code class="literal">if</code>, so
+</p><pre class="programlisting">if a=5 then a=a-1
+</pre><p>
+will be interpreted as:
+</p><pre class="programlisting">if a==5 then a:=a-1
+</pre><p>
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch05s07.html">Prev</a> </td><td width="20%" align="center">
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width="40%" align="left" valign="top">List of GEL Operators </td><td width="20%" align="center"><a
accesskey="h" href="index.html">Home</a></td><td width="40%" align="right" valign="top">
Loops</td></tr></table></div></body></html>
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Loops</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with GEL"><link rel="prev"
href="ch06.html" title="Chapter 6. Programming with GEL"><link rel="next" href="ch06s03.html" title="Sums and
Products"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Loops</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch06.html">Prev</a>
</td><th width="60%" align="center">Chapter 6. Programming with GEL</th><td width="20%" align="right"> <a
accesskey="n" href="ch06s03.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"
<a name="genius-gel-loops"></a>Loops</h2></div></div></div><div class="sect2"><div
class="titlepage"><div><div><h3 class="title"><a name="genius-gel-loops-while"></a>While
Loops</h3></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">while <expression1> do <expression2>
+until <expression1> do <expression2>
+do <expression2> while <expression1>
+do <expression2> until <expression1></pre><p>
+
+ These are similar to other languages. However, as in GEL it is simply an expression that must have
some return value, these
+ constructs will simply return the result of the last iteration or <code class="literal">NULL</code>
if no iteration was done. In the boolean expression, <code class="literal">=</code> is translated into <code
class="literal">==</code> just as for the <code class="literal">if</code> statement.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-loops-for"></a>For Loops</h3></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">for <identifier> = <from> to <to> do <body>
+for <identifier> = <from> to <to> by <increment> do <body></pre><p>
+
+Loop with identifier being set to all values from <code class="literal"><from></code> to <code
class="literal"><to></code>, optionally using an increment other than 1. These are faster, nicer and
more compact than the normal loops such as above, but less flexible. The identifier must be an identifier and
can't be a dereference. The value of identifier is the last value of identifier, or <code
class="literal"><from></code> if body was never evaluated. The variable is guaranteed to be initialized
after a loop, so you can safely use it. Also the <code class="literal"><from></code>, <code
class="literal"><to></code> and <code class="literal"><increment></code> must be non complex
values. The <code class="literal"><to></code> is not guaranteed to be hit, but will never be overshot,
for example the following prints out odd numbers from 1 to 19:
+</p><pre class="programlisting">for i = 1 to 20 by 2 do print(i)
+</pre><p>
+ </p><p>
+ When one of the values is a floating point number, then the
+ final check is done to within 2^-20 of the step size. That is,
+ even if we overshoot by 2^-20 times the "by" above, we still execute the last
+ iteration. This way
+</p><pre class="programlisting">for x = 0 to 1 by 0.1 do print(x)
+</pre><p>
+does the expected even though adding 0.1 ten times becomes just slightly more than 1.0 due to the way that
floating point numbers
+are stored in base 2 (there is no 0.1, the actual number stored is just ever so slightly bigger). This is
not perfect but it handles
+the majority of the cases. If you want to avoid dealing with this issue, use actual rational numbers for
example:
+</p><pre class="programlisting">for x = 0 to 1 by 1/10 do print(x)
+</pre><p>
+ This check is done only from version 1.0.16 onwards, so execution of your code may differ on
older versions.
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-loops-foreach"></a>Foreach Loops</h3></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">for <identifier> in <matrix> do <body></pre><p>
+
+ For each element in the matrix, going row by row from left to right we execute the
body
+ with the identifier set to the current element. To
+print numbers 1,2,3 and 4 in this order you could do:
+</p><pre class="programlisting">for n in [1,2:3,4] do print(n)
+</pre><p>
+If you wish to run through the rows and columns of a matrix, you can use
+the RowsOf and ColumnsOf functions, which return a vector of the rows or
+columns of the matrix. So,
+</p><pre class="programlisting">for n in RowsOf ([1,2:3,4]) do print(n)
+</pre><p>
+will print out [1,2] and then [3,4].
+ </p></div><div class="sect2"><div class="titlepage"><div><div><h3 class="title"><a
name="genius-gel-loops-break-continue"></a>Break and Continue</h3></div></div></div><p>
+You can also use the <code class="literal">break</code> and <code class="literal">continue</code> commands
in loops. The continue <code class="literal">continue</code> command will restart the current loop at its
next iteration, while the <code class="literal">break</code> command exits the current loop.
+</p><pre class="programlisting">while(<expression1>) do (
+ if(<expression2>) break
+ else if(<expression3>) continue;
+ <expression4>
+)
+</pre><p>
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch06.html">Prev</a> </td><td width="20%" align="center"><a
accesskey="u" href="ch06.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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diff --git a/help/C/ch06s03.html b/help/C/ch06s03.html
new file mode 100644
index 0000000..27f7589
--- /dev/null
+++ b/help/C/ch06s03.html
@@ -0,0 +1,16 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Sums and
Products</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with
GEL"><link rel="prev" href="ch06s02.html" title="Loops"><link rel="next" href="ch06s04.html"
title="Comparison Operators"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Sums and Products</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch06s02.html">Prev</a> </td><th width="60%" align="center">Chapter 6. Programming with GEL</th><td
width="20%" align="right"> <a accesskey="n" href="ch06s04.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-sums-products"></a>Sums and Products</h2></div></div></div><p>
+Syntax:
+</p><pre class="programlisting">sum <identifier> = <from> to <to> do <body>
+sum <identifier> = <from> to <to> by <increment> do <body>
+sum <identifier> in <matrix> do <body>
+prod <identifier> = <from> to <to> do <body>
+prod <identifier> = <from> to <to> by <increment> do <body>
+prod <identifier> in <matrix> do <body></pre><p>
+
+If you substitute <code class="literal">for</code> with <code class="literal">sum</code> or <code
class="literal">prod</code>, then you will get a sum or a product instead of a <code
class="literal">for</code> loop. Instead of returning the last value, these will return the sum or the
product of the values respectively.
+ </p><p>
+If no body is executed (for example <strong class="userinput"><code>sum i=1 to 0 do ...</code></strong>)
then <code class="literal">sum</code> returns 0 and <code class="literal">prod</code> returns 1 as is the
standard convention.
+ </p><p>
+ For floating point numbers the same roundoff error protection is done as in the for loop.
+ See <a class="xref" href="ch06s02.html#genius-gel-loops-for" title="For Loops">the section
called “For Loops”</a>.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch06s02.html">Prev</a> </td><td width="20%" align="center"><a
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width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Comparison Operators</td></tr></table></div></body></html>
diff --git a/help/C/ch06s04.html b/help/C/ch06s04.html
new file mode 100644
index 0000000..127f30a
--- /dev/null
+++ b/help/C/ch06s04.html
@@ -0,0 +1,40 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Comparison
Operators</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with
GEL"><link rel="prev" href="ch06s03.html" title="Sums and Products"><link rel="next" href="ch06s05.html"
title="Global Variables and Scope of Variables"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Comparison Operators</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch06s03.html">Prev</a> </td><th width="60%" align="center">Chapter 6. Programming with
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s05.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div>
<div><h2 class="title" style="clear: both"><a name="genius-gel-comparison-operators"></a>Comparison
Operators</h2></div></div></div><p>
+ The following standard comparison operators are supported in GEL and have the obvious meaning:
+ <code class="literal">==</code>, <code class="literal">>=</code>,
+ <code class="literal"><=</code>, <code class="literal">!=</code>,
+ <code class="literal"><></code>, <code class="literal"><</code>,
+ <code class="literal">></code>. They return <code class="constant">true</code> or
+ <code class="constant">false</code>.
+ The operators
+ <code class="literal">!=</code> and <code class="literal"><></code> are the same
+ thing and mean "is not equal to".
+ GEL also supports the operator
+ <code class="literal"><=></code>, which returns -1 if left side is
+ smaller, 0 if both sides are equal, 1 if left side is larger.
+ </p><p>
+ Normally <code class="literal">=</code> is translated to <code class="literal">==</code> if
+ it happens to be somewhere where GEL is expecting a condition such as
+ in the if condition. For example
+ </p><pre class="programlisting">if a=b then c
+if a==b then c
+</pre><p>
+ are the same thing in GEL. However you should really use
+ <code class="literal">==</code> or <code class="literal">:=</code> when you want to compare
+ or assign respectively if you want your code to be easy to read and
+ to avoid mistakes.
+ </p><p>
+ All the comparison operators (except for the
+ <code class="literal"><=></code> operator, which
+ behaves normally), are not strictly binary operators, they can in fact
+ be grouped in the normal mathematical way, e.g.:
+ (<code class="literal">1<x<=y<5</code>) is
+ a legal boolean expression and means just what it should, that is
+ (1<x and x≤y and y<5)
+ </p><p>
+ To build up logical expressions use the words <code class="literal">not</code>,
+ <code class="literal">and</code>, <code class="literal">or</code>, <code class="literal">xor</code>.
+ The operators <code class="literal">or</code> and <code class="literal">and</code> are
+special beasts as they evaluate their arguments one by one, so the usual trick
+for conditional evaluation works here as well. For example, <code class="literal">1 or a=1</code> will not
set
+<code class="literal">a=1</code> since the first argument was true.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch06s03.html">Prev</a> </td><td width="20%" align="center"><a
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</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
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diff --git a/help/C/ch06s05.html b/help/C/ch06s05.html
new file mode 100644
index 0000000..9a9754d
--- /dev/null
+++ b/help/C/ch06s05.html
@@ -0,0 +1,113 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Global Variables and
Scope of Variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with
GEL"><link rel="prev" href="ch06s04.html" title="Comparison Operators"><link rel="next" href="ch06s06.html"
title="Parameter variables"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Global Variables and Scope of Variables</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch06s04.html">Prev</a> </td><th width="60%" align="center">Chapter 6. Programming with
GEL</th><td width="20%" align="right"> <a accesskey="n"
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ass="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Global Variables and Scope of Variables</h2></div></div></div><p>
+ GEL is a
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ dynamically scoped language</a>. We will explain what this
+ means below. That is, normal variables and functions are dynamically
+ scoped. The exception are
+ <a class="link" href="ch06s06.html" title="Parameter variables">parameter variables</a>,
+ which are always global.
+ </p><p>
+ Like most programming languages, GEL has different types
+ of variables. Normally when a variable is defined in a function,
+ it is visible from that function and from all functions that are
+ called (all higher contexts). For example, suppose a function
+ <code class="function">f</code> defines a variable <code class="varname">a</code>
+ and then calls function <code class="function">g</code>. Then
+ function <code class="function">g</code> can reference
+ <code class="varname">a</code>. But once <code class="function">f</code> returns,
+ the variable <code class="varname">a</code> goes out of scope.
+ For example, the following code will print out 5.
+ The function <code class="function">g</code> cannot be called on the
+ top level (outside <code class="function">f</code> as <code class="varname">a</code>
+ will not be defined).
+</p><pre class="programlisting">function f() = (a:=5; g());
+function g() = print(a);
+f();
+</pre><p>
+ </p><p>
+ If you define a variable inside a function it will override
+ any variables defined in calling functions. For example,
+ we modify the above code and write:
+</p><pre class="programlisting">function f() = (a:=5; g());
+function g() = print(a);
+a:=10;
+f();
+</pre><p>
+ This code will still print out 5. But if you call
+ <code class="function">g</code> outside of <code class="function">f</code> then
+ you will get a printout of 10. Note that
+ setting <code class="varname">a</code>
+ to 5 inside <code class="function">f</code> does not change
+ the value of <code class="varname">a</code> at the top (global) level,
+ so if you now check the value of <code class="varname">a</code> it will
+ still be 10.
+ </p><p>
+ Function arguments are exactly like variables defined inside
+ the function, except that they are initialized with the value
+ that was passed to the function. Other than this point, they are
+ treated just like all other variables defined inside the
+ function.
+ </p><p>
+ Functions are treated exactly like variables. Hence you can
+ locally redefine functions. Normally (on the top level) you
+ cannot redefine protected variables and functions. But locally
+ you can do this. Consider the following session:
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>function f(x)
= sin(x)^2</code></strong>
+= (`(x)=(sin(x)^2))
+<code class="prompt">genius> </code><strong class="userinput"><code>function f(x) =
sin(x)^2</code></strong>
+= (`(x)=(sin(x)^2))
+<code class="prompt">genius> </code><strong class="userinput"><code>function g(x) = ((function
sin(x)=x^10);f(x))</code></strong>
+= (`(x)=((sin:=(`(x)=(x^10)));f(x)))
+<code class="prompt">genius> </code><strong class="userinput"><code>g(10)</code></strong>
+= 1e20
+</pre><p>
+ </p><p>
+ Functions and variables defined at the top level are
+ considered global. They are visible from anywhere. As we
+ said the following function <code class="function">f</code>
+ will not change the value of <code class="varname">a</code> to 5.
+</p><pre class="programlisting">a=6;
+function f() = (a:=5);
+f();
+</pre><p>
+ Sometimes, however, it is necessary to set
+a global variable from inside a function. When this behavior is needed,
+use the
+<a class="link" href="ch11s02.html#gel-function-set"><code class="function">set</code></a> function. Passing
a string or a quoted identifier to
+this function sets the variable globally (on the top level).
+For example, to set
+<code class="varname">a</code> to the value 3 you could call:
+</p><pre class="programlisting">set(`a,3)
+</pre><p>
+or:
+</p><pre class="programlisting">set("a",3)
+</pre><p>
+ </p><p>
+ The <code class="function">set</code> function always sets the toplevel
+ global. There is no way to set a local variable in some function
+ from a subroutine. If this is required, must use passing by
+ reference.
+ </p><p>
+ See also the
+ <a class="link" href="ch11s02.html#gel-function-SetElement"><code
class="function">SetElement</code></a> and
+ <a class="link" href="ch11s02.html#gel-function-SetVElement"><code
class="function">SetVElement</code></a> functions.
+ </p><p>
+ So to recap in a more technical language: Genius operates with
+ different numbered contexts. The top level is the context 0
+ (zero). Whenever a function is entered, the context is raised,
+ and when the function returns the context is lowered. A function
+ or a variable is always visible from all higher numbered contexts.
+ When a variable was defined in a lower numbered context, then
+ setting this variable has the effect of creating a new local
+ variable in the current context number and this variable
+ will now be visible from all higher numbered contexts.
+ </p><p>
+ There are also true local variables that are not seen from
+ anywhere but the current context. Also when returning functions
+ by value it may reference variables not visible from higher context
+ and this may be a problem. See the sections
+ <a class="link" href="ch07s04.html" title="True Local Variables">True
+ Local Variables</a> and
+ <a class="link" href="ch07s03.html" title="Returning Functions">Returning
+ Functions</a>.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch06s04.html">Prev</a> </td><td width="20%" align="center"><a
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diff --git a/help/C/ch06s06.html b/help/C/ch06s06.html
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--- /dev/null
+++ b/help/C/ch06s06.html
@@ -0,0 +1,18 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Parameter
variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with
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GEL</th><td width="20%" align="right"> <a accesskey="n"
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class="titlepage"><div><div><h2 c
lass="title" style="clear: both"><a name="genius-gel-parameters"></a>Parameter
variables</h2></div></div></div><p>
+ As we said before, there exist special variables called parameters
+ that exist in all scopes. To declare a parameter called
+ <code class="varname">foo</code> with the initial value 1, we write
+</p><pre class="programlisting">parameter foo = 1
+</pre><p>
+ From then on, <code class="varname">foo</code> is a strictly global variable.
+ Setting <code class="varname">foo</code> inside any function will modify the
+ variable in all contexts, that is, functions do not have a private
+ copy of parameters.
+ </p><p>
+ When you undefine a parameter using the
+ <a class="link" href="ch11s02.html#gel-function-undefine">
+ <code class="function">undefine</code></a> function, it stops being
+ a parameter.
+ </p><p>
+ Some parameters are built-in and modify the behavior of genius.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch06s05.html">Prev</a> </td><td width="20%" align="center"><a
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diff --git a/help/C/ch06s07.html b/help/C/ch06s07.html
new file mode 100644
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+++ b/help/C/ch06s07.html
@@ -0,0 +1,17 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Returning</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with GEL"><link rel="prev"
href="ch06s06.html" title="Parameter variables"><link rel="next" href="ch06s08.html"
title="References"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Returning</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch06s06.html">Prev</a> </td><th width="60%" align="center">Chapter 6. Programming with GEL</th><td
width="20%" align="right"> <a accesskey="n" href="ch06s08.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a nam
e="genius-gel-returning"></a>Returning</h2></div></div></div><p>
+ Normally a function is one or several expressions separated by a
+semicolon, and the value of the last expression is returned. This is fine for
+simple functions, but
+sometimes you do not want a function to return the last thing calculated. You may, for example, want to
return from a middle of a function. In this case, you can use the <code class="literal">return</code>
keyword. <code class="literal">return</code> takes one argument, which is the value to be returned.
+ </p><p>
+Example:
+</p><pre class="programlisting">function f(x) = (
+ y=1;
+ while true do (
+ if x>50 then return y;
+ y=y+1;
+ x=x+1
+ )
+)
+</pre><p>
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch06s06.html">Prev</a> </td><td width="20%" align="center"><a
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+++ b/help/C/ch06s08.html
@@ -0,0 +1,35 @@
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>References</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch06.html"
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accesskey="p" href="ch06s07.html">Prev</a> </td><th width="60%" align="center">Chapter 6. Programming with
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s09.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-g
el-references"></a>References</h2></div></div></div><p>
+ It may be necessary for some functions to return more than one value.
+ This may be accomplished by returning a vector of values, but many
+ times it is convenient to use passing a reference to a variable.
+ You pass a reference to a variable to a function, and the function
+ will set the variable for you using a dereference. You do not have
+ to use references only for this purpose, but this is their main use.
+ </p><p>
+ When using functions that return values through references
+ in the argument list, just pass the variable name with an ampersand.
+ For example the following code will compute an eigenvalue of a matrix
+ <code class="varname">A</code> with initial eigenvector guess
+ <code class="varname">x</code>, and store the computed eigenvector
+ into the variable named <code class="varname">v</code>:
+</p><pre class="programlisting">RayleighQuotientIteration (A,x,0.001,100,&v)
+</pre><p>
+ </p><p>
+The details of how references work and the syntax is similar to the C language.
+The operator
+<code class="literal">&</code> references a variable
+and <code class="literal">*</code> dereferences a variable. Both can only be applied to an identifier,
+so <code class="literal">**a</code> is not a legal expression in GEL.
+ </p><p>
+References are best explained by an example:
+</p><pre class="programlisting">a=1;
+b=&a;
+*b=2;
+</pre><p>
+now <code class="varname">a</code> contains 2. You can also reference functions:
+</p><pre class="programlisting">function f(x) = x+1;
+t=&f;
+*t(3)
+</pre><p>
+gives us 4.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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diff --git a/help/C/ch06s09.html b/help/C/ch06s09.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Lvalues</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with GEL"><link rel="prev"
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class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-lvalues"></a>Lvalues</h2></div></div></div><p>
+ An lvalue is the left hand side of an assignment. In other words, an
+ lvalue is what you assign something to. Valid lvalues are:
+</p><div class="variablelist"><dl class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a</code></strong></span></dt><dd><p>
+ Identifier. Here we would be setting the variable of name
+ <code class="varname">a</code>.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>*a</code></strong></span></dt><dd><p>
+ Dereference of an identifier. This will set whatever variable
+ <code class="varname">a</code> points to.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(<region>)</code></strong></span></dt><dd><p>
+ A region of a matrix. Here the region is specified normally as with
+ the regular @() operator, and can be a single entry, or an entire
+ region of the matrix.
+ </p></dd></dl></div><p>
+ </p><p>
+Examples:
+</p><pre class="programlisting">a:=4
+*tmp := 89
+a@(1,1) := 5
+a@(4:8,3) := [1,2,3,4,5]'
+</pre><p>
+Note that both <code class="literal">:=</code> and <code class="literal">=</code> can be used
+interchangeably. Except if the assignment appears in a condition.
+It is thus always safer to just use
+<code class="literal">:=</code> when you mean assignment, and <code class="literal">==</code>
+when you mean comparison.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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class="titlepage"><div><div><h1 class="title"><a name="genius
-gel-programming-advanced"></a>Chapter 7. Advanced Programming with GEL</h1></div></div></div><div
class="toc"><p><b>Table of Contents</b></p><dl class="toc"><dt><span class="sect1"><a
href="ch07.html#genius-gel-error-handling">Error Handling</a></span></dt><dt><span class="sect1"><a
href="ch07s02.html">Toplevel Syntax</a></span></dt><dt><span class="sect1"><a href="ch07s03.html">Returning
Functions</a></span></dt><dt><span class="sect1"><a href="ch07s04.html">True Local
Variables</a></span></dt><dt><span class="sect1"><a href="ch07s05.html">GEL Startup
Procedure</a></span></dt><dt><span class="sect1"><a href="ch07s06.html">Loading
Programs</a></span></dt></dl></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-error-handling"></a>Error Handling</h2></div></div></div><p>
+If you detect an error in your function, you can bail out of it. For normal
+errors, such as wrong types of arguments, you can fail to compute the function
+by adding the statement <code class="literal">bailout</code>. If something went
+really wrong and you want to completely kill the current computation, you can
+use <code class="literal">exception</code>.
+ </p><p>
+ For example if you want to check for arguments in your function. You
+could use the following code.
+</p><pre class="programlisting">function f(M) = (
+ if not IsMatrix (M) then (
+ error ("M not a matrix!");
+ bailout
+ );
+ ...
+)
+</pre><p>
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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diff --git a/help/C/ch07s02.html b/help/C/ch07s02.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Toplevel
Syntax</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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"><div><div><h2 class="title" style="clear: both"><a name="genius-gel-toplevel-syntax"></a>Toplevel
Syntax</h2></div></div></div><p>
+ The syntax is slightly different if you enter statements on
+ the top level versus when they are inside parentheses or
+ inside functions. On the top level, enter acts the same as if
+ you press return on the command line. Therefore think of programs
+ as just sequence of lines as if were entered on the command line.
+ In particular, you do not need to enter the separator at the end of the
+ line (unless it is of course part of several statements inside
+ parentheses).
+ </p><p>
+ The following code will produce an error when entered on the top
+ level of a program, while it will work just fine in a function.
+</p><pre class="programlisting">if Something() then
+ DoSomething()
+else
+ DoSomethingElse()
+</pre><p>
+ </p><p>
+ The problem is that after <span class="application">Genius Mathematics Tool</span> sees the end of
line after the
+ second line, it will decide that we have whole statement and
+ it will execute it. After the execution is done, <span class="application">Genius Mathematics
Tool</span> will
+ go on to the next
+ line, it will see <code class="literal">else</code>, and it will produce
+ a parsing error. To fix this, use parentheses. <span class="application">Genius Mathematics
Tool</span> will not
+ be satisfied until it has found that all parentheses are closed.
+</p><pre class="programlisting">if Something() then (
+ DoSomething()
+) else (
+ DoSomethingElse()
+)
+</pre><p>
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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diff --git a/help/C/ch07s03.html b/help/C/ch07s03.html
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index 0000000..fa74cb4
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Returning
Functions</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch07.html" title="Chapter 7. Advanced
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Programming with GEL</th><td width="20%" align="right"> <a accesskey="n"
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<h2 class="title" style="clear: both"><a name="genius-gel-returning-functions"></a>Returning
Functions</h2></div></div></div><p>
+ It is possible to return functions as value. This way you can
+ build functions that construct special purpose functions according
+ to some parameters. The tricky bit is what variables does the
+ function see. The way this works in GEL is that when a function
+ returns another function, all identifiers referenced in the
+ function body that went out of scope
+ are prepended a private dictionary of the returned
+ function. So the function will see all variables that were in
+ scope
+ when it was defined. For example, we define a function that
+ returns a function that adds 5 to its argument.
+</p><pre class="programlisting">function f() = (
+ k = 5;
+ `(x) = (x+k)
+)
+</pre><p>
+ Notice that the function adds <code class="varname">k</code> to
+ <code class="varname">x</code>. You could use this as follows.
+</p><pre class="programlisting">g = f();
+g(5)
+</pre><p>
+ And <strong class="userinput"><code>g(5)</code></strong> should return 10.
+ </p><p>
+ One thing to note is that the value of <code class="varname">k</code>
+ that is used is the one that's in effect when the
+ <code class="function">f</code> returns. For example:
+</p><pre class="programlisting">function f() = (
+ k := 5;
+ function r(x) = (x+k);
+ k := 10;
+ r
+)
+</pre><p>
+ will return a function that adds 10 to its argument rather than
+ 5. This is because the extra dictionary is created only when
+ the context
+ in which the function was defined ends, which is when the function
+ <code class="function">f</code> returns. This is consistent with how you
+ would expect the function <code class="function">r</code> to work inside
+ the function <code class="function">f</code> according to the rules of
+ scope of variables in GEL. Only those variables are added to the
+ extra dictionary that are in the context that just ended and
+ no longer exists. Variables
+ used in the function that are in still valid contexts will work
+ as usual, using the current value of the variable.
+ The only difference is with global variables and functions.
+ All identifiers that referenced global variables at time of
+ the function definition are not added to the private dictionary.
+ This is to avoid much unnecessary work when returning functions
+ and would rarely be a problem. For example, suppose that you
+ delete the "k=5" from the function <code class="function">f</code>,
+ and at the top level you define <code class="varname">k</code> to be
+ say 5. Then when you run <code class="function">f</code>, the function
+ <code class="function">r</code> will not put <code class="varname">k</code> into
+ the private dictionary because it was global (toplevel)
+ at the time of definition of <code class="function">r</code>.
+ </p><p>
+ Sometimes it is better to have more control over how variables
+ are copied into the private dictionary. Since version 1.0.7,
+ you can specify which
+ variables are copied into the private dictionary by putting
+ extra square brackets after the arguments with the list of
+ variables to be copied separated by commas.
+ If you do this, then variables are
+ copied into the private dictionary at time of the function
+ definition, and the private dictionary is not touched afterwards.
+ For example
+</p><pre class="programlisting">function f() = (
+ k := 5;
+ function r(x) [k] = (x+k);
+ k := 10;
+ r
+)
+</pre><p>
+ will return a function that when called will add 5 to its
+ argument. The local copy of <code class="varname">k</code> was created
+ when the function was defined.
+ </p><p>
+ When you want the function to not have any private dictionary
+ then put empty square brackets after the argument list. Then
+ no private dictionary will be created at all. Doing this is
+ good to increase efficiency when a private dictionary is not
+ needed or when you want the function to lookup all variables
+ as it sees them when called. For example suppose you want
+ the function returned from <code class="function">f</code> to see
+ the value of <code class="varname">k</code> from the toplevel despite
+ there being a local variable of the same name during definition.
+ So the code
+</p><pre class="programlisting">function f() = (
+ k := 5;
+ function r(x) [] = (x+k);
+ r
+);
+k := 10;
+g = f();
+g(10)
+</pre><p>
+ will return 20 and not 15, which would happen if
+ <code class="varname">k</code> with a value of 5 was added to the private
+ dictionary.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>True Local
Variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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v><div><h2 class="title" style="clear: both"><a name="genius-gel-true-local-variables"></a>True Local
Variables</h2></div></div></div><p>
+ When passing functions into other functions, the normal scoping of
+ variables might be undesired. For example:
+</p><pre class="programlisting">k := 10;
+function r(x) = (x+k);
+function f(g,x) = (
+ k := 5;
+ g(x)
+);
+f(r,1)
+</pre><p>
+ you probably want the function <code class="function">r</code>
+ when passed as <code class="function">g</code> into <code class="function">f</code>
+ to see <code class="varname">k</code> as 10 rather than 5, so that
+ the code returns 11 and not 6. However, as written, the function
+ when executed will see the <code class="varname">k</code> that is
+ equal to 5. There are two ways to solve this. One would be
+ to have <code class="function">r</code> get <code class="varname">k</code> in a
+ private dictionary using the square bracket notation section
+ <a class="link" href="ch07s03.html" title="Returning Functions">Returning
+ Functions</a>.
+ </p><p>
+ But there is another solution. Since version 1.0.7 there are
+ true local variables. These are variables that are visible only
+ from the current context and not from any called functions.
+ We could define <code class="varname">k</code> as a local variable in the
+ function <code class="function">f</code>. To do this add a
+ <span class="command"><strong>local</strong></span> statement as the first statement in the
+ function (it must always be the first statement in the function).
+ You can also make any arguments be local variables as well.
+ That is,
+</p><pre class="programlisting">function f(g,x) = (
+ local g,x,k;
+ k := 5;
+ g(x)
+);
+</pre><p>
+ Then the code will work as expected and prints out 11.
+ Note that the <span class="command"><strong>local</strong></span> statement initializes
+ all the referenced variables (except for function arguments) to
+ a <code class="constant">null</code>.
+ </p><p>
+ If all variables are to be created as locals you can just pass an
+ asterisk instead of a list of variables. In this case the variables
+ will not be initialized until they are actually set of course.
+ So the following definition of <code class="function">f</code>
+ will also work:
+</p><pre class="programlisting">function f(g,x) = (
+ local *;
+ k := 5;
+ g(x)
+);
+</pre><p>
+ </p><p>
+ It is good practice that all functions that take other functions
+ as arguments use local variables. This way the passed function
+ does not see implementation details and get confused.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>GEL Startup
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<div><h2 class="title" style="clear: both"><a name="genius-gel-startup-procedure"></a>GEL Startup
Procedure</h2></div></div></div><p>
+First the program looks for the installed library file (the compiled version <code
class="filename">lib.cgel</code>) in the installed directory, then it looks into the current directory, and
then it tries to load an uncompiled file called
+<code class="filename">~/.geniusinit</code>.
+ </p><p>
+If you ever change the library in its installed place, you’ll have to
+first compile it with <span class="command"><strong>genius --compile loader.gel > lib.cgel</strong></span>
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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diff --git a/help/C/ch07s06.html b/help/C/ch07s06.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Loading
Programs</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch07.html" title="Chapter 7. Advanced
Programming with GEL"><link rel="prev" href="ch07s05.html" title="GEL Startup Procedure"><link rel="next"
href="ch08.html" title="Chapter 8. Matrices in GEL"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Loading Programs</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch07s05.html">Prev</a> </td><th width="60%" align="center">Chapter 7. Advanced
Programming with GEL</th><td width="20%" align="right"> <a accesskey="n"
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v><h2 class="title" style="clear: both"><a name="genius-gel-loading-programs"></a>Loading
Programs</h2></div></div></div><p>
+Sometimes you have a larger program you wrote into a file and want to read that file into <span
class="application">Genius Mathematics Tool</span>. In these situations, you have two options. You can keep
the functions you use most inside the <code class="filename">~/.geniusinit</code> file. Or if you want to
load up a file in a middle of a session (or from within another file), you can type <span
class="command"><strong>load <list of filenames></strong></span> at the prompt. This has to be done on
the top level and not inside any function or whatnot, and it cannot be part of any expression. It also has a
slightly different syntax than the rest of genius, more similar to a shell. You can enter the file in quotes.
If you use the '' quotes, you will get exactly the string that you typed, if you use the "" quotes, special
characters will be unescaped as they are for strings. Example:
+</p><pre class="programlisting">load program1.gel program2.gel
+load "Weird File Name With SPACES.gel"
+</pre><p>
+There are also <span class="command"><strong>cd</strong></span>, <span
class="command"><strong>pwd</strong></span> and <span class="command"><strong>ls</strong></span> commands
built in. <span class="command"><strong>cd</strong></span> will take one argument, <span
class="command"><strong>ls</strong></span> will take an argument that is like the glob in the UNIX shell
(i.e., you can use wildcards). <span class="command"><strong>pwd</strong></span> takes no arguments. For
example:
+</p><pre class="programlisting">cd directory_with_gel_programs
+ls *.gel
+</pre><p>
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch07s05.html">Prev</a> </td><td width="20%" align="center"><a
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diff --git a/help/C/ch08.html b/help/C/ch08.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 8. Matrices in
GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="index.html" title="Genius Manual"><link
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class="titlepage"><div><div><h1 class="title"><a name
="genius-gel-matrices"></a>Chapter 8. Matrices in GEL</h1></div></div></div><div class="toc"><p><b>Table of
Contents</b></p><dl class="toc"><dt><span class="sect1"><a
href="ch08.html#genius-gel-matrix-support">Entering Matrices</a></span></dt><dt><span class="sect1"><a
href="ch08s02.html">Conjugate Transpose and Transpose Operator</a></span></dt><dt><span class="sect1"><a
href="ch08s03.html">Linear Algebra</a></span></dt></dl></div><p>
+ Genius has support for vectors and matrices and possesses a sizable library of
+ matrix manipulation and linear algebra functions.
+ </p><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-matrix-support"></a>Entering Matrices</h2></div></div></div><p>
+To enter matrices, you can use one of the following two syntaxes. You can either enter
+the matrix on one line, separating values by commas and rows by semicolons. Or you
+can enter each row on one line, separating
+values by commas.
+You can also just combine the two methods.
+So to enter a 3x3 matrix
+of numbers 1-9 you could do
+</p><pre class="programlisting">[1,2,3;4,5,6;7,8,9]
+</pre><p>
+or
+</p><pre class="programlisting">[1, 2, 3
+ 4, 5, 6
+ 7, 8, 9]
+</pre><p>
+Do not use both ';' and return at once on the same line though.
+ </p><p>
+You can also use the matrix expansion functionality to enter matrices.
+For example you can do:
+</p><pre class="programlisting">a = [ 1, 2, 3
+ 4, 5, 6
+ 7, 8, 9]
+b = [ a, 10
+ 11, 12]
+</pre><p>
+and you should get
+</p><pre class="programlisting">[1, 2, 3, 10
+ 4, 5, 6, 10
+ 7, 8, 9, 10
+ 11, 11, 11, 12]
+</pre><p>
+similarly you can build matrices out of vectors and other stuff like that.
+ </p><p>
+Another thing is that non-specified spots are initialized to 0, so
+</p><pre class="programlisting">[1, 2, 3
+ 4, 5
+ 6]
+</pre><p>
+will end up being
+</p><pre class="programlisting">
+[1, 2, 3
+ 4, 5, 0
+ 6, 0, 0]
+</pre><p>
+ </p><p>
+ When matrices are evaluated, they are evaluated and traversed row-wise. This is just
+ like the <code class="literal">M@(j)</code> operator, which traverses the matrix row-wise.
+ </p><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Note</h3><p>
+Be careful about using returns for expressions inside the
+<code class="literal">[ ]</code> brackets, as they have a slightly different meaning
+there. You will start a new row.
+ </p></div></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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diff --git a/help/C/ch08s02.html b/help/C/ch08s02.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Conjugate Transpose
and Transpose Operator</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch08.html" title="Chapter 8. Matrices
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align="center">Chapter 8. Matrices in GEL</th><td width="20%" align="right"> <a accesskey="n"
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tlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-gel-matrix-transpose"></a>Conjugate
Transpose and Transpose Operator</h2></div></div></div><p>
+You can conjugate transpose a matrix by using the <code class="literal">'</code> operator. That is
+the entry in the
+<code class="varname">i</code>th column and the <code class="varname">j</code>th row will be
+the complex conjugate of the entry in the
+<code class="varname">j</code>th column and the <code class="varname">i</code>th row of the original matrix.
+ For example:
+</p><pre class="programlisting">[1,2,3]*[4,5,6]'
+</pre><p>
+We transpose the second vector to make matrix multiplication possible.
+If you just want to transpose a matrix without conjugating it, you would
+use the <code class="literal">.'</code> operator. For example:
+</p><pre class="programlisting">[1,2,3]*[4,5,6i].'
+</pre><p>
+ </p><p>
+ Note that normal transpose, that is the <code class="literal">.'</code> operator, is much faster
+ and will not create a new copy of the matrix in memory. The conjugate transpose does
+ create a new copy unfortunately.
+ It is recommended to always use the <code class="literal">.'</code> operator when working with real
+ matrices and vectors.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch08.html">Prev</a> </td><td width="20%" align="center"><a
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diff --git a/help/C/ch08s03.html b/help/C/ch08s03.html
new file mode 100644
index 0000000..ebe1591
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Linear
Algebra</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch08.html" title="Chapter 8. Matrices in
GEL"><link rel="prev" href="ch08s02.html" title="Conjugate Transpose and Transpose Operator"><link rel="next"
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header"><tr><th colspan="3" align="center">Linear Algebra</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch08s02.html">Prev</a> </td><th width="60%" align="center">Chapter 8. Matrices in
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch09.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
cl
ass="title" style="clear: both"><a name="genius-gel-matrix-linalg"></a>Linear
Algebra</h2></div></div></div><p>
+ Genius implements many useful linear algebra and matrix manipulation
+routines. See the <a class="link" href="ch11s09.html" title="Linear Algebra">Linear Algebra</a> and
+<a class="link" href="ch11s08.html" title="Matrix Manipulation">Matrix Manipulation</a>
+sections of the GEL function listing.
+ </p><p>
+ The linear algebra routines implemented in GEL do not currently come
+from a well tested numerical package, and thus should not be used for critical
+numerical computation. On the other hand, Genius implements very well many
+linear algebra operations with rational and integer coefficients. These are
+inherently exact and in fact will give you much better results than common
+double precision routines for linear algebra.
+ </p><p>
+ For example, it is pointless to compute the rank and nullspace of a
+floating point matrix since for all practical purposes, we need to consider the
+matrix as having some slight errors. You are likely to get a different result
+than you expect. The problem is that under a small perturbation every matrix
+is of full rank and invertible. If the matrix however is of rational numbers,
+then the rank and nullspace are always exact.
+ </p><p>
+ In general when Genius computes the basis of a certain vectorspace
+ (for example with the <a class="link" href="ch11s09.html#gel-function-NullSpace"><code
class="function">NullSpace</code></a>) it will give the basis as
+a matrix, in which the columns are the vectors of the basis. That is, when
+Genius talks of a linear subspace it means a matrix whose column space is
+the given linear subspace.
+ </p><p>
+ It should be noted that Genius can remember certain properties of a
+matrix. For example, it will remember that a matrix is in row reduced form.
+If many calls are made to functions that internally use row reduced form of
+the matrix, we can just row reduce the matrix beforehand once. Successive
+calls to <a class="link" href="ch11s09.html#gel-function-rref"><code class="function">rref</code></a> will
be very fast.
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
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diff --git a/help/C/ch09.html b/help/C/ch09.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 9. Polynomials
in GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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accesskey="n" href="ch10.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-gel-
polynomials"></a>Chapter 9. Polynomials in GEL</h1></div></div></div><div class="toc"><p><b>Table of
Contents</b></p><dl class="toc"><dt><span class="sect1"><a
href="ch09.html#genius-gel-polynomials-using">Using Polynomials</a></span></dt></dl></div><p>
+ Currently Genius can handle polynomials of one variable written out
+ as vectors, and do some basic operations with these. It is planned to
+ expand this support further.
+ </p><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-polynomials-using"></a>Using Polynomials</h2></div></div></div><p>
+Currently
+polynomials in one variable are just horizontal vectors with value only nodes.
+The power of the term is the position in the vector, with the first position
+being 0. So,
+</p><pre class="programlisting">[1,2,3]
+</pre><p>
+translates to a polynomial of
+</p><pre class="programlisting">1 + 2*x + 3*x^2
+</pre><p>
+ </p><p>
+You can add, subtract and multiply polynomials using the
+<a class="link" href="ch11s15.html#gel-function-AddPoly"><code class="function">AddPoly</code></a>,
+<a class="link" href="ch11s15.html#gel-function-SubtractPoly"><code
class="function">SubtractPoly</code></a>, and
+<a class="link" href="ch11s15.html#gel-function-MultiplyPoly"><code class="function">MultiplyPoly</code></a>
functions respectively.
+You can print a polynomial using the
+<a class="link" href="ch11s15.html#gel-function-PolyToString"><code class="function">PolyToString</code></a>
+function.
+For example,
+</p><pre class="programlisting">PolyToString([1,2,3],"y")
+</pre><p>
+gives
+</p><pre class="programlisting">3*y^2 + 2*y + 1
+</pre><p>
+You can also get a function representation of the polynomial so that you can
+evaluate it. This is done by using
+<a class="link" href="ch11s15.html#gel-function-PolyToFunction"><code
class="function">PolyToFunction</code></a>,
+which
+returns an anonymous function.
+</p><pre class="programlisting">f = PolyToFunction([0,1,1])
+f(2)
+</pre><p>
+ </p><p>
+ It is also possible to find roots of polynomials of degrees 1 through 4 by using the
+function
+<a class="link" href="ch11s13.html#gel-function-PolynomialRoots"><code
class="function">PolynomialRoots</code></a>,
+which calls the appropriate formula function. Higher degree polynomials must be converted to
+functions and solved
+numerically using a function such as
+<a class="link" href="ch11s13.html#gel-function-FindRootBisection"><code
class="function">FindRootBisection</code></a>,
+<a class="link" href="ch11s13.html#gel-function-FindRootFalsePosition"><code
class="function">FindRootFalsePosition</code></a>,
+<a class="link" href="ch11s13.html#gel-function-FindRootMullersMethod"><code
class="function">FindRootMullersMethod</code></a>, or
+<a class="link" href="ch11s13.html#gel-function-FindRootSecant"><code
class="function">FindRootSecant</code></a>.
+ </p><p>
+See <a class="xref" href="ch11s15.html" title="Polynomials">the section called “Polynomials”</a> in the
function list
+for the rest of functions acting on polynomials.
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch08s03.html">Prev</a> </td><td width="20%" align="center">
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href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Chapter 10. Set Theory in
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diff --git a/help/C/ch10.html b/help/C/ch10.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 10. Set Theory
in GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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class="chapter"><div class="titlepage"><div><div><h1 class="title"><a na
me="genius-gel-settheory"></a>Chapter 10. Set Theory in GEL</h1></div></div></div><div
class="toc"><p><b>Table of Contents</b></p><dl class="toc"><dt><span class="sect1"><a
href="ch10.html#genius-gel-sets-using">Using Sets</a></span></dt></dl></div><p>
+ Genius has some basic set theoretic functionality built in. Currently a set is
+ just a vector (or a matrix). Every distinct object is treated as a different element.
+ </p><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-sets-using"></a>Using Sets</h2></div></div></div><p>
+ Just like vectors, objects
+ in sets can include numbers, strings, <code class="constant">null</code>, matrices and vectors. It is
+ planned in the future to have a dedicated type for sets, rather than using vectors.
+ Note that floating point numbers are distinct from integers, even if they appear the same.
+ That is, Genius will treat <code class="constant">0</code> and <code class="constant">0.0</code>
+ as two distinct elements. The <code class="constant">null</code> is treated as an empty set.
+ </p><p>
+ To build a set out of a vector, use the
+ <a class="link" href="ch11s16.html#gel-function-MakeSet"><code class="function">MakeSet</code></a>
function.
+ Currently, it will just return a new vector where every element is unique.
+</p><pre class="screen"><code class="prompt">genius> </code><strong
class="userinput"><code>MakeSet([1,2,2,3])</code></strong>
+= [1, 2, 3]
+</pre><p>
+</p><p>
+ Similarly there are functions
+ <a class="link" href="ch11s16.html#gel-function-Union"><code class="function">Union</code></a>,
+ <a class="link" href="ch11s16.html#gel-function-Intersection"><code
class="function">Intersection</code></a>,
+ <a class="link" href="ch11s16.html#gel-function-SetMinus"><code class="function">SetMinus</code></a>,
which
+ are rather self explanatory. For example:
+</p><pre class="screen"><code class="prompt">genius> </code><strong
class="userinput"><code>Union([1,2,3], [1,2,4])</code></strong>
+= [1, 2, 4, 3]
+</pre><p>
+ Note that no order is guaranteed for the return values. If you wish to sort the vector you
+should use the
+ <a class="link" href="ch11s08.html#gel-function-SortVector"><code
class="function">SortVector</code></a> function.
+ </p><p>
+ For testing membership, there are functions
+ <a class="link" href="ch11s16.html#gel-function-IsIn"><code class="function">IsIn</code></a> and
+ <a class="link" href="ch11s16.html#gel-function-IsSubset"><code class="function">IsSubset</code></a>,
+ which return a boolean value. For example:
+</p><pre class="screen"><code class="prompt">genius> </code><strong class="userinput"><code>IsIn (1,
[0,1,2])</code></strong>
+= true
+</pre><p>
+ The input <strong class="userinput"><code>IsIn(x,X)</code></strong> is of course equivalent to
+ <strong class="userinput"><code>IsSubset([x],X)</code></strong>. Note that since the empty set is a
subset
+ of every set, <strong class="userinput"><code>IsSubset(null,X)</code></strong> is always true.
+ </p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td
width="40%" align="left"><a accesskey="p" href="ch09.html">Prev</a> </td><td width="20%" align="center">
</td><td width="40%" align="right"> <a accesskey="n" href="ch11.html">Next</a></td></tr><tr><td width="40%"
align="left" valign="top">Chapter 9. Polynomials in GEL </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Chapter 11. List of GEL
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 11. List of
GEL functions</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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class="titlepage"><div><div><h1 class="title"><a name="genius-gel-f
unction-list"></a>Chapter 11. List of GEL functions</h1></div></div></div><div class="toc"><p><b>Table of
Contents</b></p><dl class="toc"><dt><span class="sect1"><a
href="ch11.html#genius-gel-function-list-commands">Commands</a></span></dt><dt><span class="sect1"><a
href="ch11s02.html">Basic</a></span></dt><dt><span class="sect1"><a
href="ch11s03.html">Parameters</a></span></dt><dt><span class="sect1"><a
href="ch11s04.html">Constants</a></span></dt><dt><span class="sect1"><a
href="ch11s05.html">Numeric</a></span></dt><dt><span class="sect1"><a
href="ch11s06.html">Trigonometry</a></span></dt><dt><span class="sect1"><a href="ch11s07.html">Number
Theory</a></span></dt><dt><span class="sect1"><a href="ch11s08.html">Matrix
Manipulation</a></span></dt><dt><span class="sect1"><a href="ch11s09.html">Linear
Algebra</a></span></dt><dt><span class="sect1"><a href="ch11s10.html">Combinatorics</a></span></dt><dt><span
class="sect1"><a href="ch11s11.html">Calculus</a></span></dt><dt><sp
an class="sect1"><a href="ch11s12.html">Functions</a></span></dt><dt><span class="sect1"><a
href="ch11s13.html">Equation Solving</a></span></dt><dt><span class="sect1"><a
href="ch11s14.html">Statistics</a></span></dt><dt><span class="sect1"><a
href="ch11s15.html">Polynomials</a></span></dt><dt><span class="sect1"><a href="ch11s16.html">Set
Theory</a></span></dt><dt><span class="sect1"><a href="ch11s17.html">Commutative
Algebra</a></span></dt><dt><span class="sect1"><a href="ch11s18.html">Miscellaneous</a></span></dt><dt><span
class="sect1"><a href="ch11s19.html">Symbolic Operations</a></span></dt><dt><span class="sect1"><a
href="ch11s20.html">Plotting</a></span></dt></dl></div><p>
+To get help on a specific function from the console type:
+</p><pre class="programlisting">help FunctionName
+</pre><p>
+ </p><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-commands"></a>Commands</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-command-help"></a>help</span></dt><dd><pre
class="synopsis">help</pre><pre class="synopsis">help FunctionName</pre><p>Print help (or help on a
function/command).</p></dd><dt><span class="term"><a name="gel-command-load"></a>load</span></dt><dd><pre
class="synopsis">load "file.gel"</pre><p>Load a file into the interpreter. The file will execute
+as if it were typed onto the command line.</p></dd><dt><span class="term"><a
name="gel-command-cd"></a>cd</span></dt><dd><pre class="synopsis">cd /directory/name</pre><p>Change working
directory to <code class="filename">/directory/name</code>.</p></dd><dt><span class="term"><a
name="gel-command-pwd"></a>pwd</span></dt><dd><pre class="synopsis">pwd</pre><p>Print the current working
directory.</p></dd><dt><span class="term"><a name="gel-command-ls"></a>ls</span></dt><dd><pre
class="synopsis">ls</pre><p>List files in the current directory.</p></dd><dt><span class="term"><a
name="gel-command-plugin"></a>plugin</span></dt><dd><pre class="synopsis">plugin plugin_name</pre><p>Load a
plugin. Plugin of that name must be installed on the system
+in the proper directory.</p></dd></dl></div></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch10.html">Prev</a>
</td><td width="20%" align="center"> </td><td width="40%" align="right"> <a accesskey="n"
href="ch11s02.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Chapter 10. Set Theory in
GEL </td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Basic</td></tr></table></div></body></html>
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Basic</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
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alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Basic</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch11.html">Prev</a>
</td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td width="20%" align="right"> <a
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class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="genius-gel-function-list-basic"></a>Basic</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-AskButtons"></a>AskButtons</span></dt><dd><pre class="synopsis">AskButtons
(query)</pre><pre class="synopsis">AskButtons (query, button1, ...)</pre><p>Asks a question and presents a
list of buttons to the user (or
+a menu of options in text mode). Returns the 1-based index of the button
+pressed. That is, returns 1 if the first button was pressed, 2 if the second
+button was pressed, and so on. If the user closes the window (or simply hits
+enter in text mode), then <code class="constant">null</code> is returned. The execution
+of the program is blocked until the user responds.</p><p>Version 1.0.10 onwards.</p></dd><dt><span
class="term"><a name="gel-function-AskString"></a>AskString</span></dt><dd><pre class="synopsis">AskString
(query)</pre><pre class="synopsis">AskString (query, default)</pre><p>Asks a question and lets the user enter
a string, which
+it then returns. If the user cancels or closes the window, then
+<code class="constant">null</code> is returned. The execution of the program
+is blocked until the user responds. If <code class="varname">default</code> is given, then it is pre-typed
in for the user to just press enter on (version 1.0.6 onwards).</p></dd><dt><span class="term"><a
name="gel-function-Compose"></a>Compose</span></dt><dd><pre class="synopsis">Compose (f,g)</pre><p>Compose
two functions and return a function that is the composition of <code class="function">f</code> and <code
class="function">g</code>.</p></dd><dt><span class="term"><a
name="gel-function-ComposePower"></a>ComposePower</span></dt><dd><pre class="synopsis">ComposePower
(f,n,x)</pre><p>Compose and execute a function with itself <code class="varname">n</code> times, passing
<code class="varname">x</code> as argument. Returning <code class="varname">x</code> if
+<code class="varname">n</code> equals 0.
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>function f(x) = x^2 ;</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>ComposePower (f,3,7)</code></strong>
+= 5764801
+<code class="prompt">genius></code> <strong class="userinput"><code>f(f(f(7)))</code></strong>
+= 5764801
+</pre><p>
+ </p></dd><dt><span class="term"><a name="gel-function-Evaluate"></a>Evaluate</span></dt><dd><pre
class="synopsis">Evaluate (str)</pre><p>Parses and evaluates a string.</p></dd><dt><span class="term"><a
name="gel-function-GetCurrentModulo"></a>GetCurrentModulo</span></dt><dd><pre
class="synopsis">GetCurrentModulo</pre><p>Get current modulo from the context outside the function. That is,
if outside of
+the function was executed in modulo (using <code class="literal">mod</code>) then this returns what
+this modulo was. Normally the body of the function called is not executed in modular arithmetic,
+and this builtin function makes it possible to make GEL functions aware of modular
arithmetic.</p></dd><dt><span class="term"><a name="gel-function-Identity"></a>Identity</span></dt><dd><pre
class="synopsis">Identity (x)</pre><p>Identity function, returns its argument. It is equivalent to <strong
class="userinput"><code>function Identity(x)=x</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-IntegerFromBoolean"></a>IntegerFromBoolean</span></dt><dd><pre
class="synopsis">IntegerFromBoolean (bval)</pre><p>
+ Make integer (0 for <code class="constant">false</code> or 1 for
+ <code class="constant">true</code>) from a boolean value.
+ <code class="varname">bval</code> can also be a number in which case a
+ non-zero value will be interpreted as <code class="constant">true</code> and
+ zero will be interpreted as <code class="constant">false</code>.
+ </p></dd><dt><span class="term"><a name="gel-function-IsBoolean"></a>IsBoolean</span></dt><dd><pre
class="synopsis">IsBoolean (arg)</pre><p>Check if argument is a boolean (and not a number).</p></dd><dt><span
class="term"><a name="gel-function-IsDefined"></a>IsDefined</span></dt><dd><pre class="synopsis">IsDefined
(id)</pre><p>Check if an id is defined. You should pass a string or
+ and identifier. If you pass a matrix, each entry will be
+ evaluated separately and the matrix should contain strings
+ or identifiers.</p></dd><dt><span class="term"><a
name="gel-function-IsFunction"></a>IsFunction</span></dt><dd><pre class="synopsis">IsFunction
(arg)</pre><p>Check if argument is a function.</p></dd><dt><span class="term"><a
name="gel-function-IsFunctionOrIdentifier"></a>IsFunctionOrIdentifier</span></dt><dd><pre
class="synopsis">IsFunctionOrIdentifier (arg)</pre><p>Check if argument is a function or an
identifier.</p></dd><dt><span class="term"><a
name="gel-function-IsFunctionRef"></a>IsFunctionRef</span></dt><dd><pre class="synopsis">IsFunctionRef
(arg)</pre><p>Check if argument is a function reference. This includes variable
+references.</p></dd><dt><span class="term"><a name="gel-function-IsMatrix"></a>IsMatrix</span></dt><dd><pre
class="synopsis">IsMatrix (arg)</pre><p>Check if argument is a matrix. Even though <code
class="constant">null</code> is
+sometimes considered an empty matrix, the function <code class="function">IsMatrix</code> does
+not consider <code class="constant">null</code> a matrix.</p></dd><dt><span class="term"><a
name="gel-function-IsNull"></a>IsNull</span></dt><dd><pre class="synopsis">IsNull (arg)</pre><p>Check if
argument is a <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsString"></a>IsString</span></dt><dd><pre class="synopsis">IsString (arg)</pre><p>Check
if argument is a text string.</p></dd><dt><span class="term"><a
name="gel-function-IsValue"></a>IsValue</span></dt><dd><pre class="synopsis">IsValue (arg)</pre><p>Check if
argument is a number.</p></dd><dt><span class="term"><a
name="gel-function-Parse"></a>Parse</span></dt><dd><pre class="synopsis">Parse (str)</pre><p>Parses but does
not evaluate a string. Note that certain
+ pre-computation is done during the parsing stage.</p></dd><dt><span class="term"><a
name="gel-function-SetFunctionFlags"></a>SetFunctionFlags</span></dt><dd><pre
class="synopsis">SetFunctionFlags (id,flags...)</pre><p>Set flags for a function, currently <code
class="literal">"PropagateMod"</code> and <code class="literal">"NoModuloArguments"</code>.
+If <code class="literal">"PropagateMod"</code> is set, then the body of the function is evaluated in modular
arithmetic when the function
+is called inside a block that was evaluated using modular arithmetic (using <code
class="literal">mod</code>). If
+<code class="literal">"NoModuloArguments"</code>, then the arguments of the function are never evaluated
using modular arithmetic.
+ </p></dd><dt><span class="term"><a name="gel-function-SetHelp"></a>SetHelp</span></dt><dd><pre
class="synopsis">SetHelp (id,category,desc)</pre><p>Set the category and help description line for a
function.</p></dd><dt><span class="term"><a
name="gel-function-SetHelpAlias"></a>SetHelpAlias</span></dt><dd><pre class="synopsis">SetHelpAlias
(id,alias)</pre><p>Sets up a help alias.</p></dd><dt><span class="term"><a
name="gel-function-chdir"></a>chdir</span></dt><dd><pre class="synopsis">chdir (dir)</pre><p>Changes current
directory, same as the <span class="command"><strong>cd</strong></span>.</p></dd><dt><span class="term"><a
name="gel-function-CurrentTime"></a>CurrentTime</span></dt><dd><pre
class="synopsis">CurrentTime</pre><p>Returns the current UNIX time with microsecond precision as a floating
point number. That is, returns the number of seconds since January 1st 1970.</p><p>Version 1.0.15
onwards.</p></dd><dt><span class="term"><a name="gel-function-display"></a>display
</span></dt><dd><pre class="synopsis">display (str,expr)</pre><p>Display a string and an expression with a
colon to separate them.</p></dd><dt><span class="term"><a
name="gel-function-DisplayVariables"></a>DisplayVariables</span></dt><dd><pre
class="synopsis">DisplayVariables (var1,var2,...)</pre><p>Display set of variables. The variables can be
given as
+ strings or identifiers. For example:
+ </p><pre class="programlisting">DisplayVariables(`x,`y,`z)
+ </pre><p>
+ </p><p>
+ If called without arguments (must supply empty argument list) as
+ </p><pre class="programlisting">DisplayVariables()
+ </pre><p>
+ then all variables are printed including a stacktrace similar to
+ <span class="guilabel">Show user variables</span> in the graphical version.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-error"></a>error</span></dt><dd><pre class="synopsis">error (str)</pre><p>Prints a string
to the error stream (onto the console).</p></dd><dt><span class="term"><a
name="gel-function-exit"></a>exit</span></dt><dd><pre class="synopsis">exit</pre><p>Aliases: <code
class="function">quit</code></p><p>Exits the program.</p></dd><dt><span class="term"><a
name="gel-function-false"></a>false</span></dt><dd><pre class="synopsis">false</pre><p>Aliases: <code
class="function">False</code> <code class="function">FALSE</code></p><p>The <code
class="constant">false</code> boolean value.</p></dd><dt><span class="term"><a
name="gel-function-manual"></a>manual</span></dt><dd><pre class="synopsis">manual</pre><p>Displays the user
manual.</p></dd><dt><span class="term"><a name="gel-function-print"></a>print</span></dt><dd><pre
class="synopsis">print (str)</pre><p>Prints an expression and then print a newline
. The argument <code class="varname">str</code> can be any expression. It is
+made into a string before being printed.</p></dd><dt><span class="term"><a
name="gel-function-printn"></a>printn</span></dt><dd><pre class="synopsis">printn (str)</pre><p>Prints an
expression without a trailing newline. The argument <code class="varname">str</code> can be any expression.
It is
+made into a string before being printed.</p></dd><dt><span class="term"><a
name="gel-function-PrintTable"></a>PrintTable</span></dt><dd><pre class="synopsis">PrintTable
(f,v)</pre><p>Print a table of values for a function. The values are in the
+ vector <code class="varname">v</code>. You can use the vector
+ building notation as follows:
+ </p><pre class="programlisting">PrintTable (f,[0:10])
+ </pre><p>
+ If <code class="varname">v</code> is a positive integer, then the table of
+ integers from 1 up to and including v will be used.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-protect"></a>protect</span></dt><dd><pre class="synopsis">protect (id)</pre><p>Protect a
variable from being modified. This is used on the internal GEL functions to
+avoid them being accidentally overridden.</p></dd><dt><span class="term"><a
name="gel-function-ProtectAll"></a>ProtectAll</span></dt><dd><pre class="synopsis">ProtectAll
()</pre><p>Protect all currently defined variables, parameters and
+functions from being modified. This is used on the internal GEL functions to
+avoid them being accidentally overridden. Normally <span class="application">Genius Mathematics Tool</span>
considers
+unprotected variables as user defined.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-set"></a>set</span></dt><dd><pre class="synopsis">set (id,val)</pre><p>Set a global
variable. The <code class="varname">id</code>
+ can be either a string or a quoted identifier.
+ For example:
+ </p><pre class="programlisting">set(`x,1)
+ </pre><p>
+ will set the global variable <code class="varname">x</code> to the value 1.
+ </p><p>The function returns the <code class="varname">val</code>, to be
+ usable in chaining.</p></dd><dt><span class="term"><a
name="gel-function-SetElement"></a>SetElement</span></dt><dd><pre class="synopsis">SetElement
(id,row,col,val)</pre><p>Set an element of a global variable which is a matrix.
+ The <code class="varname">id</code>
+ can be either a string or a quoted identifier.
+ For example:
+ </p><pre class="programlisting">SetElement(`x,2,3,1)
+ </pre><p>
+ will set the second row third column element of the global variable <code
class="varname">x</code> to the value 1. If no global variable of the name exists, or if it is set to
something that's not a matrix, a new zero matrix of appropriate size will be created.
+ </p><p>The <code class="varname">row</code> and <code class="varname">col</code> can also be
ranges, and the semantics are the same as for regular setting of the elements with an equals sign.
+ </p><p>The function returns the <code class="varname">val</code>, to be
+ usable in chaining.</p><p>Available from 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SetVElement"></a>SetVElement</span></dt><dd><pre class="synopsis">SetElement
(id,elt,val)</pre><p>Set an element of a global variable which is a vector.
+ The <code class="varname">id</code>
+ can be either a string or a quoted identifier.
+ For example:
+ </p><pre class="programlisting">SetElement(`x,2,1)
+ </pre><p>
+ will set the second element of the global vector variable <code class="varname">x</code> to the
value 1. If no global variable of the name exists, or if it is set to something that's not a vector
(matrix), a new zero row vector of appropriate size will be created.
+ </p><p>The <code class="varname">elt</code> can also be a range, and the semantics are the same as
for regular setting of the elements with an equals sign.
+ </p><p>The function returns the <code class="varname">val</code>, to be
+ usable in chaining.</p><p>Available from 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-string"></a>string</span></dt><dd><pre class="synopsis">string (s)</pre><p>Make a string.
This will make a string out of any argument.</p></dd><dt><span class="term"><a
name="gel-function-true"></a>true</span></dt><dd><pre class="synopsis">true</pre><p>Aliases: <code
class="function">True</code> <code class="function">TRUE</code></p><p>The <code class="constant">true</code>
boolean value.</p></dd><dt><span class="term"><a
name="gel-function-undefine"></a>undefine</span></dt><dd><pre class="synopsis">undefine (id)</pre><p>Alias:
<code class="function">Undefine</code></p><p>Undefine a variable. This includes locals and globals,
+ every value on all context levels is wiped. This function
+ should really not be used on local variables. A vector of
+ identifiers can also be passed to undefine several variables.
+ </p></dd><dt><span class="term"><a
name="gel-function-UndefineAll"></a>UndefineAll</span></dt><dd><pre class="synopsis">UndefineAll
()</pre><p>Undefine all unprotected global variables
+ (including functions and parameters). Normally <span class="application">Genius Mathematics
Tool</span>
+ considers protected variables as system defined functions
+ and variables. Note that <code class="function">UndefineAll</code>
+ only removes the global definition of symbols not local ones,
+ so that it may be run from inside other functions safely.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-unprotect"></a>unprotect</span></dt><dd><pre class="synopsis">unprotect
(id)</pre><p>Unprotect a variable from being modified.</p></dd><dt><span class="term"><a
name="gel-function-UserVariables"></a>UserVariables</span></dt><dd><pre class="synopsis">UserVariables
()</pre><p>Return a vector of identifiers of
+ user defined (unprotected) global variables.</p><p>Version 1.0.7 onwards.</p></dd><dt><span
class="term"><a name="gel-function-wait"></a>wait</span></dt><dd><pre class="synopsis">wait
(secs)</pre><p>Waits a specified number of seconds. <code class="varname">secs</code>
+must be non-negative. Zero is accepted and nothing happens in this case,
+except possibly user interface events are processed.</p><p>Since version 1.0.18, <code
class="varname">secs</code> can be a noninteger number, so
+ <strong class="userinput"><code>wait(0.1)</code></strong> will wait for one tenth of a
second.</p></dd><dt><span class="term"><a name="gel-function-version"></a>version</span></dt><dd><pre
class="synopsis">version</pre><p>Returns the version of Genius as a horizontal 3-vector with
+ major version first, then minor version and finally the patch level.</p></dd><dt><span
class="term"><a name="gel-function-warranty"></a>warranty</span></dt><dd><pre
class="synopsis">warranty</pre><p>Gives the warranty information.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s03.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Chapter 11. List of GEL
functions </td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Parameters</td></tr></table></div></body></html>
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charset=UTF-8"><title>Parameters</title><meta name="generator" content="DocBook XSL Stylesheets
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header"><tr><th colspan="3" align="center">Parameters</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s02.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s04.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius
-gel-function-parameters"></a>Parameters</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ChopTolerance"></a>ChopTolerance</span></dt><dd><pre class="synopsis">ChopTolerance =
number</pre><p>Tolerance of the <code class="function">Chop</code> function.</p></dd><dt><span
class="term"><a name="gel-function-ContinuousNumberOfTries"></a>ContinuousNumberOfTries</span></dt><dd><pre
class="synopsis">ContinuousNumberOfTries = number</pre><p>How many iterations to try to find the limit for
continuity and limits.</p></dd><dt><span class="term"><a
name="gel-function-ContinuousSFS"></a>ContinuousSFS</span></dt><dd><pre class="synopsis">ContinuousSFS =
number</pre><p>How many successive steps to be within tolerance for calculation of
continuity.</p></dd><dt><span class="term"><a
name="gel-function-ContinuousTolerance"></a>ContinuousTolerance</span></dt><dd><pre
class="synopsis">ContinuousTolerance = number</pre><p>Toler
ance for continuity of functions and for calculating the limit.</p></dd><dt><span class="term"><a
name="gel-function-DerivativeNumberOfTries"></a>DerivativeNumberOfTries</span></dt><dd><pre
class="synopsis">DerivativeNumberOfTries = number</pre><p>How many iterations to try to find the limit for
derivative.</p></dd><dt><span class="term"><a
name="gel-function-DerivativeSFS"></a>DerivativeSFS</span></dt><dd><pre class="synopsis">DerivativeSFS =
number</pre><p>How many successive steps to be within tolerance for calculation of
derivative.</p></dd><dt><span class="term"><a
name="gel-function-DerivativeTolerance"></a>DerivativeTolerance</span></dt><dd><pre
class="synopsis">DerivativeTolerance = number</pre><p>Tolerance for calculating the derivatives of
functions.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunctionTolerance"></a>ErrorFunctionTolerance</span></dt><dd><pre
class="synopsis">ErrorFunctionTolerance = number</pre><p>Tolerance of the <a class="link" href
="ch11s12.html#gel-function-ErrorFunction"><code
class="function">ErrorFunction</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-FloatPrecision"></a>FloatPrecision</span></dt><dd><pre class="synopsis">FloatPrecision =
number</pre><p>Floating point precision.</p></dd><dt><span class="term"><a
name="gel-function-FullExpressions"></a>FullExpressions</span></dt><dd><pre class="synopsis">FullExpressions
= boolean</pre><p>Print full expressions, even if more than a line.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistributionTolerance"></a>GaussDistributionTolerance</span></dt><dd><pre
class="synopsis">GaussDistributionTolerance = number</pre><p>Tolerance of the <a class="link"
href="ch11s14.html#gel-function-GaussDistribution"><code class="function">GaussDistribution</code></a>
function.</p></dd><dt><span class="term"><a
name="gel-function-IntegerOutputBase"></a>IntegerOutputBase</span></dt><dd><pre
class="synopsis">IntegerOutputBase = number</pre><
p>Integer output base.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimeMillerRabinReps"></a>IsPrimeMillerRabinReps</span></dt><dd><pre
class="synopsis">IsPrimeMillerRabinReps = number</pre><p>Number of extra Miller-Rabin tests to run on a
number before declaring it a prime in <a class="link" href="ch11s07.html#gel-function-IsPrime"><code
class="function">IsPrime</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawLegends"></a>LinePlotDrawLegends</span></dt><dd><pre
class="synopsis">LinePlotDrawLegends = true</pre><p>Tells genius to draw the legends for <a class="link"
href="ch11s20.html" title="Plotting">line plotting
+ functions</a> such as <a class="link" href="ch11s20.html#gel-function-LinePlot"><code
class="function">LinePlot</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawAxisLabels"></a>LinePlotDrawAxisLabels</span></dt><dd><pre
class="synopsis">LinePlotDrawAxisLabels = true</pre><p>Tells genius to draw the axis labels for <a
class="link" href="ch11s20.html" title="Plotting">line plotting
+ functions</a> such as <a class="link" href="ch11s20.html#gel-function-LinePlot"><code
class="function">LinePlot</code></a>.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotVariableNames"></a>LinePlotVariableNames</span></dt><dd><pre
class="synopsis">LinePlotVariableNames = ["x","y","z","t"]</pre><p>Tells genius which variable names are used
as default names for <a class="link" href="ch11s20.html" title="Plotting">line plotting
+ functions</a> such as <a class="link" href="ch11s20.html#gel-function-LinePlot"><code
class="function">LinePlot</code></a> and friends.
+ </p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotWindow"></a>LinePlotWindow</span></dt><dd><pre class="synopsis">LinePlotWindow =
[x1,x2,y1,y2]</pre><p>Sets the limits for <a class="link" href="ch11s20.html" title="Plotting">line plotting
+ functions</a> such as <a class="link" href="ch11s20.html#gel-function-LinePlot"><code
class="function">LinePlot</code></a>.
+ </p></dd><dt><span class="term"><a name="gel-function-MaxDigits"></a>MaxDigits</span></dt><dd><pre
class="synopsis">MaxDigits = number</pre><p>Maximum digits to display.</p></dd><dt><span class="term"><a
name="gel-function-MaxErrors"></a>MaxErrors</span></dt><dd><pre class="synopsis">MaxErrors =
number</pre><p>Maximum errors to display.</p></dd><dt><span class="term"><a
name="gel-function-MixedFractions"></a>MixedFractions</span></dt><dd><pre class="synopsis">MixedFractions =
boolean</pre><p>If true, mixed fractions are printed.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegralFunction"></a>NumericalIntegralFunction</span></dt><dd><pre
class="synopsis">NumericalIntegralFunction = function</pre><p>The function used for numerical integration in
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegralSteps"></a>Nume
ricalIntegralSteps</span></dt><dd><pre class="synopsis">NumericalIntegralSteps = number</pre><p>Steps to
perform in <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-OutputChopExponent"></a>OutputChopExponent</span></dt><dd><pre
class="synopsis">OutputChopExponent = number</pre><p>When another number in the object being printed (a
matrix or a
+value) is greater than
+10<sup>-OutputChopWhenExponent</sup>, and
+the number being printed is less than
+10<sup>-OutputChopExponent</sup>, then
+display <code class="computeroutput">0.0</code> instead of the number.
+</p><p>
+Output is never chopped if <code class="function">OutputChopExponent</code> is zero.
+It must be a non-negative integer.
+</p><p>
+If you want output always chopped according to
+<code class="function">OutputChopExponent</code>, then set
+<code class="function">OutputChopWhenExponent</code>, to something
+greater than or equal to
+<code class="function">OutputChopExponent</code>.
+</p></dd><dt><span class="term"><a
name="gel-function-OutputChopWhenExponent"></a>OutputChopWhenExponent</span></dt><dd><pre
class="synopsis">OutputChopWhenExponent = number</pre><p>When to chop output. See
+ <a class="link" href="ch11s03.html#gel-function-OutputChopExponent"><code
class="function">OutputChopExponent</code></a>.
+</p></dd><dt><span class="term"><a name="gel-function-OutputStyle"></a>OutputStyle</span></dt><dd><pre
class="synopsis">OutputStyle = string</pre><p>
+ Output style, this can be <code class="literal">normal</code>, <code
class="literal">latex</code>, <code class="literal">mathml</code> or <code class="literal">troff</code>.
+ </p><p>
+ This affects mostly how matrices and fractions are printed out and
+ is useful for pasting into documents. For example you can set this
+ to the latex by:
+ </p><pre class="programlisting">OutputStyle = "latex"
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-ResultsAsFloats"></a>ResultsAsFloats</span></dt><dd><pre class="synopsis">ResultsAsFloats
= boolean</pre><p>Convert all results to floats before printing.</p></dd><dt><span class="term"><a
name="gel-function-ScientificNotation"></a>ScientificNotation</span></dt><dd><pre
class="synopsis">ScientificNotation = boolean</pre><p>Use scientific notation.</p></dd><dt><span
class="term"><a name="gel-function-SlopefieldTicks"></a>SlopefieldTicks</span></dt><dd><pre
class="synopsis">SlopefieldTicks = [vertical,horizontal]</pre><p>Sets the number of vertical and horizontal
ticks in a
+slopefield plot. (See <a class="link" href="ch11s20.html#gel-function-SlopefieldPlot"><code
class="function">SlopefieldPlot</code></a>).
+ </p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SumProductNumberOfTries"></a>SumProductNumberOfTries</span></dt><dd><pre
class="synopsis">SumProductNumberOfTries = number</pre><p>How many iterations to try for <a class="link"
href="ch11s11.html#gel-function-InfiniteSum"><code class="function">InfiniteSum</code></a> and <a
class="link" href="ch11s11.html#gel-function-InfiniteProduct"><code
class="function">InfiniteProduct</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SumProductSFS"></a>SumProductSFS</span></dt><dd><pre class="synopsis">SumProductSFS =
number</pre><p>How many successive steps to be within tolerance for <a class="link"
href="ch11s11.html#gel-function-InfiniteSum"><code class="function">InfiniteSum</code></a> and <a
class="link" href="ch11s11.html#gel-function-InfiniteProduct"><code
class="function">InfiniteProduct</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SumProductTolerance
"></a>SumProductTolerance</span></dt><dd><pre class="synopsis">SumProductTolerance =
number</pre><p>Tolerance for <a class="link" href="ch11s11.html#gel-function-InfiniteSum"><code
class="function">InfiniteSum</code></a> and <a class="link"
href="ch11s11.html#gel-function-InfiniteProduct"><code
class="function">InfiniteProduct</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLegends"></a>SurfacePlotDrawLegends</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLegends = true</pre><p>Tells genius to draw the legends for <a class="link"
href="ch11s20.html" title="Plotting">surface plotting
+ functions</a> such as <a class="link" href="ch11s20.html#gel-function-SurfacePlot"><code
class="function">SurfacePlot</code></a>.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotVariableNames"></a>SurfacePlotVariableNames</span></dt><dd><pre
class="synopsis">SurfacePlotVariableNames = ["x","y","z"]</pre><p>Tells genius which variable names are used
as default names for <a class="link" href="ch11s20.html" title="Plotting">surface plotting
+ functions</a> using <a class="link" href="ch11s20.html#gel-function-SurfacePlot"><code
class="function">SurfacePlot</code></a>.
+ Note that the <code class="varname">z</code> does not refer to the dependent (vertical) axis, but
to the independent complex variable
+ <strong class="userinput"><code>z=x+iy</code></strong>.
+ </p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotWindow"></a>SurfacePlotWindow</span></dt><dd><pre
class="synopsis">SurfacePlotWindow = [x1,x2,y1,y2,z1,z2]</pre><p>Sets the limits for surface plotting (See <a
class="link" href="ch11s20.html#gel-function-SurfacePlot"><code
class="function">SurfacePlot</code></a>).</p></dd><dt><span class="term"><a
name="gel-function-VectorfieldNormalized"></a>VectorfieldNormalized</span></dt><dd><pre
class="synopsis">VectorfieldNormalized = true</pre><p>Should the vectorfield plotting have normalized arrow
length. If true, vector fields will only show direction
+ and not magnitude. (See <a class="link" href="ch11s20.html#gel-function-VectorfieldPlot"><code
class="function">VectorfieldPlot</code></a>).
+ </p></dd><dt><span class="term"><a
name="gel-function-VectorfieldTicks"></a>VectorfieldTicks</span></dt><dd><pre
class="synopsis">VectorfieldTicks = [vertical,horizontal]</pre><p>Sets the number of vertical and horizontal
ticks in a
+vectorfield plot. (See <a class="link" href="ch11s20.html#gel-function-VectorfieldPlot"><code
class="function">VectorfieldPlot</code></a>).
+ </p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s02.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s04.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Basic </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Constants</td></tr></table></div></body></html>
diff --git a/help/C/ch11s04.html b/help/C/ch11s04.html
new file mode 100644
index 0000000..af54505
--- /dev/null
+++ b/help/C/ch11s04.html
@@ -0,0 +1,44 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Constants</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
href="ch11s03.html" title="Parameters"><link rel="next" href="ch11s05.html" title="Numeric"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Constants</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s03.html">Prev</a> </td><th width="60%"
align="center">Chapter 11. List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="geniu
s-gel-function-list-constants"></a>Constants</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>
+ Catalan's Constant, approximately 0.915... It is defined to be the series where terms are
<strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, where <code class="varname">k</code>
ranges from 0 to infinity.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Aliases: <code class="function">gamma</code></p><p>
+ Euler's constant gamma. Sometimes called the
+ Euler-Mascheroni constant.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>The
Golden Ratio.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
+ in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
+ round and uniform.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
+ The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
+ is the exponential function
+ <a class="link" href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a>. It
is approximately
+ 2.71828182846... This number is sometimes called Euler's number, although there are
+ several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>
+ The number pi, that is the ratio of a circle's circumference
+ to its diameter. This is approximately 3.14159265359...
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Parameters </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Numeric</td></tr></table></div></body></html>
diff --git a/help/C/ch11s05.html b/help/C/ch11s05.html
new file mode 100644
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--- /dev/null
+++ b/help/C/ch11s05.html
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Numeric</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
href="ch11s04.html" title="Constants"><link rel="next" href="ch11s06.html" title="Trigonometry"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Numeric</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s04.html">Prev</a> </td><th width="60%"
align="center">Chapter 11. List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="geniu
s-gel-function-list-numeric"></a>Numeric</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Aliases: <code class="function">abs</code></p><p>
+ Absolute value of a number and if <code class="varname">x</code> is
+ a complex value the modulus of <code class="varname">x</code>. I.e. this
+ the distance of <code class="varname">x</code> to the origin. This is equivalent
+ to <strong class="userinput"><code>|x|</code></strong>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
+ <a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld
(complex modulus)</a>
+for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Replace very small number with zero.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Aliases: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calculates the complex conjugate of the complex number <code
class="varname">z</code>. If <code class="varname">z</code> is a vector or matrix,
+all its elements are conjugated.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Get the denominator of a rational number.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Return the fractional part of a number.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Aliases: <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Division without remainder.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
+ <strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
+ <strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Check if argument is a possibly complex rational number.
That is, if both real and imaginary parts are
+ given as rational numbers. Of course rational simply means "not stored as a floating point
number."</p></dd><dt><span class="term"><a name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre
class="synopsis">IsFloat (num)</pre><p>Check if argument is a real floating point number
(non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(num)</pre><p>Aliases: <code class="function">IsComplexInteger</code></p><p>Check if argument is a possibly
complex integer. That is a complex integer is a number of
+ the form <strong class="userinput"><code>n+1i*m</code></strong> where <code
class="varname">n</code> and <code class="varname">m</code>
+ are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Check if argument is an integer (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Aliases: <code
class="function">IsNaturalNumber</code></p><p>Check if argument is a positive real integer. Note that
+we accept the convention that 0 is not a natural number.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Check if argument is a rational number (non-complex). Of course rational simply means "not
stored as a floating point number."</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (num)</pre><p>Check if
argument is a real number.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Get
the numerator of a rational number.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Aliases: <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Aliases: <code class="function">sign</code></p><p>Return the sign of a
number. That is returns
+<code class="literal">-1</code> if value is negative,
+<code class="literal">0</code> if value is zero and
+<code class="literal">1</code> if value is positive. If <code class="varname">x</code> is a complex
+value then <code class="function">Sign</code> returns the direction or 0.
+ </p></dd><dt><span class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre
class="synopsis">ceil (x)</pre><p>Aliases: <code class="function">Ceiling</code></p><p>Get the lowest integer
more than or equal to <code class="varname">n</code>. Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ceil(1.1)</code></strong>
+= 2
+<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1.1)</code></strong>
+= -1
+</pre><p>
+ </p><p>Note that you should be careful and notice that floating point
+ numbers are stored in binary and so may not be what you
+ expect. For example <strong class="userinput"><code>ceil(420/4.2)</code></strong>
+ returns 101 instead of the expected 100. This is because
+ 4.2 is actually very slightly less than 4.2. Use rational
+ representation <strong class="userinput"><code>42/10</code></strong> if you want
+ exact arithmetic.
+ </p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>
+ The exponential function. This is the function
+ <strong class="userinput"><code>e^x</code></strong> where <code class="varname">e</code>
+ is the <a class="link" href="ch11s04.html#gel-function-e">base of the natural
+ logarithm</a>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Make number a floating point value. That is returns the floating point
representation of the number <code class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Aliases: <code
class="function">Floor</code></p><p>Get the highest integer less than or equal to <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>The natural logarithm, the logarithm to base <code
class="varname">e</code>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logarithm of <code
class="varname">x</code> base <code class="varname">b</code> (calls <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> if in modulo
mode), if base is not given, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> is used.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logarithm of <code
class="varname">x</code> base 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Aliases: <code
class="function">lg</code></p><p>Logarithm of <code class="varname">x</code> base 2.</p></dd><dt><span
class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synop
sis">max (a,args...)</pre><p>Aliases: <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Returns the maximum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,args...)</pre><p>Aliases: <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Returns the minimum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(size...)</pre><p>Generate random float in the range <code class="literal">[0,1)</code>.
+If size is given then a matrix (if two numbers are specified) or vector (if one
+number is specified) of the given size returned.</p></dd><dt><span class="term"><a
name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,size...)</pre><p>Generate random integer in the range
+<code class="literal">[0,max)</code>.
+If size is given then a matrix (if two numbers are specified) or vector (if one
+number is specified) of the given size returned. For example,
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
+= 3
+<code class="prompt">genius></code> <strong class="userinput"><code>randint(4,2)</code></strong>
+=
+[0 1]
+<code class="prompt">genius></code> <strong class="userinput"><code>randint(4,2,3)</code></strong>
+=
+[2 2 1
+ 0 0 3]
+</pre><p>
+ </p></dd><dt><span class="term"><a name="gel-function-round"></a>round</span></dt><dd><pre
class="synopsis">round (x)</pre><p>Aliases: <code class="function">Round</code></p><p>Round a
number.</p></dd><dt><span class="term"><a name="gel-function-sqrt"></a>sqrt</span></dt><dd><pre
class="synopsis">sqrt (x)</pre><p>Aliases: <code class="function">SquareRoot</code></p><p>The square root.
When operating modulo some integer will return either a <code class="constant">null</code> or a vector of the
square roots. Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>sqrt(2)</code></strong>
+= 1.41421356237
+<code class="prompt">genius></code> <strong class="userinput"><code>sqrt(-1)</code></strong>
+= 1i
+<code class="prompt">genius></code> <strong class="userinput"><code>sqrt(4) mod 7</code></strong>
+=
+[2 5]
+<code class="prompt">genius></code> <strong class="userinput"><code>2*2 mod 7</code></strong>
+= 4
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Square_root"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/SquareRoot"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-trunc"></a>trunc</span></dt><dd><pre
class="synopsis">trunc (x)</pre><p>Aliases: <code class="function">Truncate</code> <code
class="function">IntegerPart</code></p><p>Truncate number to an integer (return the integer
part).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s04.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s06.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Constants </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Trigonometry</td></tr></table></div></body></html>
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new file mode 100644
index 0000000..71fe61e
--- /dev/null
+++ b/help/C/ch11s06.html
@@ -0,0 +1,64 @@
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometry</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s05.html" title="Numeric"><link
rel="next" href="ch11s07.html" title="Number Theory"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometry</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a na
me="genius-gel-function-list-trigonometry"></a>Trigonometry</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Aliases: <code
class="function">arccos</code></p><p>The arccos (inverse cos) function.</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Aliases: <code
class="function">arccosh</code></p><p>The arccosh (inverse cosh) function.</p></dd><dt><span class="term"><a
name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Aliases: <code
class="function">arccot</code></p><p>The arccot (inverse cot) function.</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Aliases: <code
class="function">arccoth</code></p><p>The arccoth (inverse coth) function.</p></dd><dt><spa
n class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc
(x)</pre><p>Aliases: <code class="function">arccsc</code></p><p>The inverse cosecant
function.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Aliases: <code class="function">arccsch</code></p><p>The inverse
hyperbolic cosecant function.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Aliases: <code
class="function">arcsec</code></p><p>The inverse secant function.</p></dd><dt><span class="term"><a
name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech (x)</pre><p>Aliases: <code
class="function">arcsech</code></p><p>The inverse hyperbolic secant function.</p></dd><dt><span
class="term"><a name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin
(x)</pre><p>Aliases: <code class="function">arcsi
n</code></p><p>The arcsin (inverse sin) function.</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Aliases: <code
class="function">arcsinh</code></p><p>The arcsinh (inverse sinh) function.</p></dd><dt><span class="term"><a
name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Aliases: <code
class="function">arctan</code></p><p>Calculates the arctan (inverse tan) function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Aliases: <code class="function">arctanh</code></p><p>The arctanh (inverse
tanh) function.</p></dd><dt><span class="term"><a name="gel-function-atan2"></a>atan2</span></dt><dd><pre
class="synopsis">atan2 (y, x)</pre><p>Aliases: <code class="function">arctan2</code></p><p>Calculates the
arctan2 function. If
+ <strong class="userinput"><code>x>0</code></strong> then it returns
+ <strong class="userinput"><code>atan(y/x)</code></strong>. If <strong
class="userinput"><code>x<0</code></strong>
+ then it returns <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>.
+ When <strong class="userinput"><code>x=0</code></strong> it returns <strong
class="userinput"><code>sign(y) *
+ pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> returns 0
+ rather than failing.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Calculates the cosine function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Calculates the hyperbolic cosine function.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>The cotangent function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>The hyperbolic cotangent function.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>The cosecant function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>The hyperbolic cosecant function.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>The secant function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>The hyperbolic secant function.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Calculates the sine function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Calculates the hyperbolic sine function.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calculates the tan function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>The hyperbolic tangent function.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s05.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s07.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Numeric </td><td
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diff --git a/help/C/ch11s07.html b/help/C/ch11s07.html
new file mode 100644
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Number
Theory</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL
functions"><link rel="prev" href="ch11s06.html" title="Trigonometry"><link rel="next" href="ch11s08.html"
title="Matrix Manipulation"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Number Theory</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s06.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td
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class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear
: both"><a name="genius-gel-function-list-number-theory"></a>Number Theory</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>
+ Are the real integers <code class="varname">a</code> and <code class="varname">b</code>
relatively prime?
+ Returns <code class="constant">true</code> or <code class="constant">false</code>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coprime_integers";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Return the <code class="varname">n</code>th Bernoulli number.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Aliases: <code class="function">CRT</code></p><p>Find the
<code class="varname">x</code> that solves the system given by
+ the vector <code class="varname">a</code> and modulo the elements of
+ <code class="varname">m</code>, using the Chinese Remainder Theorem.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Given two factorizations, give the factorization of the
+ product.</p><p>See <a class="link"
href="ch11s07.html#gel-function-Factorize">Factorize</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Convert a vector of values indicating powers of b to a number.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Convert a number to a vector of powers for elements in base
<code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>Find discrete log of <code class="varname">n</code> base <code class="varname">b</code> in
+ F<sub>q</sub>, the finite field of order <code class="varname">q</code>, where <code
class="varname">q</code>
+ is a prime, using the Silver-Pohlig-Hellman algorithm.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Checks divisibility (if <code class="varname">m</code> divides <code
class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>
+ Compute the Euler phi function for <code class="varname">n</code>, that is
+ the number of integers between 1 and <code class="varname">n</code>
+ relatively prime to <code class="varname">n</code>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>
+ Return <strong class="userinput"><code>n/d</code></strong> but only if <code
class="varname">d</code>
+ divides <code class="varname">n</code>. If <code class="varname">d</code>
+ does not divide <code class="varname">n</code> then this function returns
+ garbage. This is a lot faster for very large numbers
+ than the operation <strong class="userinput"><code>n/d</code></strong>, but of course only
+ useful if you know that the division is exact.
+ </p></dd><dt><span class="term"><a name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre
class="synopsis">Factorize (n)</pre><p>
+ Return factorization of a number as a matrix. The first
+ row is the primes in the factorization (including 1) and the
+ second row are the powers. So for example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>Factorize(11*11*13)</code></strong>
+=
+[1 11 13
+ 1 2 1]</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>
+ Return all factors of <code class="varname">n</code> in a vector. This
+ includes all the non-prime factors as well. It includes 1 and the
+ number itself. So for example to print all the perfect numbers
+ (those that are sums of their factors) up to the number 1000 you
+ could do (this is of course very inefficient)
+ </p><pre class="programlisting">for n=1 to 1000 do (
+ if MatrixSum (Factors(n)) == 2*n then
+ print(n)
+)
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,tries)</pre><p>
+ Attempt Fermat factorization of <code class="varname">n</code> into
+ <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, returns <code
class="varname">t</code>
+ and <code class="varname">s</code> as a vector if possible, <code class="constant">null</code>
otherwise.
+ <code class="varname">tries</code> specifies the number of tries before
+ giving up.
+ </p><p>
+ This is a fairly good factorization if your number is the product
+ of two factors that are very close to each other.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Find the first primitive element in F<sub>q</sub>, the
finite
+group of order <code class="varname">q</code>. Of course <code class="varname">q</code> must be a
prime.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Find a random primitive element in F<sub>q</sub>,
the finite
+group of order <code class="varname">q</code> (q must be a prime).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of n in F<sub>q</sub>, the finite
+group of order <code class="varname">q</code> (<code class="varname">q</code> a prime), using the
+factor base <code class="varname">S</code>. <code class="varname">S</code> should be a column of
+primes possibly with second column precalculated by
+<a class="link" href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Run the precalculation step of
+ <a class="link" href="ch11s07.html#gel-function-IndexCalculus"><code
class="function">IndexCalculus</code></a> for logarithms base <code class="varname">b</code> in
+F<sub>q</sub>, the finite group of order <code class="varname">q</code>
+(<code class="varname">q</code> a prime), for the factor base <code class="varname">S</code> (where
+<code class="varname">S</code> is a column vector of primes). The logs will be
+precalculated and returned in the second column.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven (n)</pre><p>Tests if an
integer is even.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>
+ Tests if a positive integer <code class="varname">p</code> is a
+ Mersenne prime exponent. That is if
+ 2<sup>p</sup>-1 is a prime. It does this
+ by looking it up in a table of known values, which is relatively
+ short.
+ See also
+ <a class="link" href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a>
+ and
+ <a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
+ for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower (m,n)</pre><p>
+ Tests if a rational number <code class="varname">m</code> is a perfect
+ <code class="varname">n</code>th power. See also
+ <a class="link" href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a>
+ and
+ <a class="link" href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.
+ </p></dd><dt><span class="term"><a name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre
class="synopsis">IsOdd (n)</pre><p>Tests if an integer is odd.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
+ Check an integer for being a perfect square of an integer. The number must
+ be a real integer. Negative integers are of course never perfect
+ squares of real integers.
+ </p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>
+ Tests primality of integers, for numbers less than 2.5e10 the
+ answer is deterministic (if Riemann hypothesis is true). For
+ numbers larger, the probability of a false positive
+ depends on
+ <a class="link" href="ch11s03.html#gel-function-IsPrimeMillerRabinReps">
+ <code class="function">IsPrimeMillerRabinReps</code></a>. That
+ is the probability of false positive is 1/4 to the power
+ <code class="function">IsPrimeMillerRabinReps</code>. The default
+ value of 22 yields a probability of about 5.7e-14.
+ </p><p>
+ If <code class="constant">false</code> is returned, you can be sure that
+ the number is a composite. If you want to be absolutely sure
+ that you have a prime you can use
+ <a class="link" href="ch11s07.html#gel-function-MillerRabinTestSure">
+ <code class="function">MillerRabinTestSure</code></a> but it may take
+ a lot longer.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre class="synopsis">IsPrimitiveMod
(g,q)</pre><p>Check if <code class="varname">g</code> is primitive in F<sub>q</sub>, the finite
+group of order <code class="varname">q</code>, where <code class="varname">q</code> is a prime. If <code
class="varname">q</code> is not prime results are bogus.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimitiveModWithPrimeFactors</span></dt><dd><pre
class="synopsis">IsPrimitiveModWithPrimeFactors (g,q,f)</pre><p>Check if <code class="varname">g</code> is
primitive in F<sub>q</sub>, the finite
+group of order <code class="varname">q</code>, where <code class="varname">q</code> is a prime and
+<code class="varname">f</code> is a vector of prime factors of <code class="varname">q</code>-1.
+If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><span class="term"><a
name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre class="synopsis">IsPseudoprime
(n,b)</pre><p>If <code class="varname">n</code> is a pseudoprime base <code class="varname">b</code> but not
a prime,
+ that is if <strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong>. This calls
the <a class="link" href="ch11s07.html#gel-function-PseudoprimeTest"><code
class="function">PseudoprimeTest</code></a></p></dd><dt><span class="term"><a
name="gel-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Test if <code class="varname">n</code> is a strong
pseudoprime to base <code class="varname">b</code> but not a prime.</p></dd><dt><span class="term"><a
name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi (a,b)</pre><p>Aliases:
<code class="function">JacobiSymbol</code></p><p>Calculate the Jacobi symbol (a/b) (b should be
odd).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Aliases: <code class="function">JacobiKroneckerSymbol</code></p><p>Calculate the Jacobi s
ymbol (a/b) with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.</p></dd><dt><span
class="term"><a name="gel-function-LeastAbsoluteResidue"></a>LeastAbsoluteResidue</span></dt><dd><pre
class="synopsis">LeastAbsoluteResidue (a,n)</pre><p>Return the residue of <code class="varname">a</code> mod
<code class="varname">n</code> with the least absolute value (in the interval -n/2 to n/2).</p></dd><dt><span
class="term"><a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre class="synopsis">Legendre
(a,p)</pre><p>Aliases: <code class="function">LegendreSymbol</code></p><p>Calculate the Legendre symbol
(a/p).</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/LegendreSymbol"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Test if 2<sup>p</sup>-1 is a Mersenne prime using the Lucas-Lehmer test.
+ See also
+ <a class="link" href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a>
+ and
+ <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returns the <code class="varname">n</code>th Lucas number.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Return all maximal prime power factors of a
number.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>
+ A vector of known Mersenne prime exponents, that is
+ a list of positive integers
+ <code class="varname">p</code> such that
+ 2<sup>p</sup>-1 is a prime.
+ See also
+ <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>
+ and
+ <a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
+ for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre class="synopsis">MillerRabinTest
(n,reps)</pre><p>
+ Use the Miller-Rabin primality test on <code class="varname">n</code>,
+ <code class="varname">reps</code> number of times. The probability of false
+ positive is <strong class="userinput"><code>(1/4)^reps</code></strong>. It is probably
+ usually better to use
+ <a class="link" href="ch11s07.html#gel-function-IsPrime">
+ <code class="function">IsPrime</code></a> since that is faster and
+ better on smaller integers.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
+ Use the Miller-Rabin primality test on <code class="varname">n</code> with
+ enough bases that assuming the Generalized Riemann Hypothesis the
+ result is deterministic.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
+ <a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Returns inverse of n mod m.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/ModularInverse.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre
class="synopsis">MoebiusMu (n)</pre><p>
+ Return the Moebius mu function evaluated in <code class="varname">n</code>.
+ That is, it returns 0 if <code class="varname">n</code> is not a product
+ of distinct primes and <strong class="userinput"><code>(-1)^k</code></strong> if it is
+ a product of <code class="varname">k</code> distinct primes.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/MoebiusFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre
class="synopsis">NextPrime (n)</pre><p>
+ Returns the least prime greater than <code class="varname">n</code>.
+ Negatives of primes are considered prime and so to get the
+ previous prime you can use <strong class="userinput"><code>-NextPrime(-n)</code></strong>.
+ </p><p>
+ This function uses the GMPs <code class="function">mpz_nextprime</code>,
+ which in turn uses the probabilistic Miller-Rabin test
+ (See also <a class="link" href="ch11s07.html#gel-function-MillerRabinTest"><code
class="function">MillerRabinTest</code></a>).
+ The probability
+ of false positive is not tunable, but is low enough
+ for all practical purposes.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synopsis">PadicValuation
(n,p)</pre><p>Returns the p-adic valuation (number of trailing zeros in base <code
class="varname">p</code>).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/P-adic_order"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/PAdicValuation"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre
class="synopsis">PowerMod (a,b,m)</pre><p>
+ Compute <strong class="userinput"><code>a^b mod m</code></strong>. The
+ <code class="varname">b</code>'s power of <code class="varname">a</code> modulo
+ <code class="varname">m</code>. It is not necessary to use this function
+ as it is automatically used in modulo mode. Hence
+ <strong class="userinput"><code>a^b mod m</code></strong> is just as fast.
+ </p></dd><dt><span class="term"><a name="gel-function-Prime"></a>Prime</span></dt><dd><pre
class="synopsis">Prime (n)</pre><p>Aliases: <code class="function">prime</code></p><p>Return the <code
class="varname">n</code>th prime (up to a limit).</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre class="synopsis">PrimeFactors
(n)</pre><p>Return all prime factors of a number as a vector.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Prime_factor"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">PseudoprimeTest
(n,b)</pre><p>Pseudoprime test, returns <code class="constant">true</code> if and only if
+ <strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong></p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Pseudoprime"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Pseudoprime.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre class="synopsis">RemoveFactor
(n,m)</pre><p>Removes all instances of the factor <code class="varname">m</code> from the number <code
class="varname">n</code>. That is divides by the largest power of <code class="varname">m</code>, that
divides <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Divisibility"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Factor.html"; target="_top">Mathworld</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Find discrete log of <code
class="varname">n</code> base <code class="varname">b</code> in F<sub>q</sub>, the finite group of order
<code class="varname">q</code>, where <code class="varname">q</code> is a prime using the
Silver-Pohlig-Hellman algorithm, given <code class="varname">f</code> being the factorization of <code
class="varname">q</code>-1.</p></dd><dt><span class="term"><a
name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Find square root of <code class="varname">n</code> modulo <code class="varname">p</code> (where
<code class="varname">p</code> is a prime). Null is returned if not a quadratic residue.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuadraticResidue"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticResidue.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Run the strong pseudoprime test base <code
class="varname">b</code> on <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Strong_pseudoprime";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/StrongPseudoprime"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/StrongPseudoprime.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-gcd"></a>gcd</span></dt><dd><pre
class="synopsis">gcd (a,args...)</pre><p>Aliases: <code class="function">GCD</code></p><p>
+ Greatest common divisor of integers. You can enter as many
+ integers as you want in the argument list, or you can give
+ a vector or a matrix of integers. If you give more than
+ one matrix of the same size then GCD is done element by
+ element.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greatest_common_divisor";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmath</a>,
or
+ <a class="ulink" href="http://mathworld.wolfram.com/GreatestCommonDivisor.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-lcm"></a>lcm</span></dt><dd><pre
class="synopsis">lcm (a,args...)</pre><p>Aliases: <code class="function">LCM</code></p><p>
+ Least common multiplier of integers. You can enter as many
+ integers as you want in the argument list, or you can give a
+ vector or a matrix of integers. If you give more than one
+ matrix of the same size then LCM is done element by element.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Least_common_multiple";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LeastCommonMultiple"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/LeastCommonMultiple.html";
target="_top">Mathworld</a> for more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="sect1"><div class="titlepage"><div><div><h2 class="title" styl
e="clear: both"><a name="genius-gel-function-list-matrix"></a>Matrix Manipulation</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Apply a function over all entries of a matrix and return a matrix of the
results.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Apply a function over all entries of 2 matrices (or 1
value and 1 matrix) and return a matrix of the results.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Gets
the columns of a matrix as a horizontal vector.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre class="synopsis">
ComplementSubmatrix (m,r,c)</pre><p>Remove column(s) and row(s) from a matrix.</p></dd><dt><span
class="term"><a name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre
class="synopsis">CompoundMatrix (k,A)</pre><p>Calculate the kth compound matrix of A.</p></dd><dt><span
class="term"><a name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
+ Count the number of zero columns in a matrix. For example
+ once your column reduce a matrix you can use this to find
+ the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
+ and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Delete a column of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Delete a row of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Gets the diagonal entries of a matrix as a column vector.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
+ same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Dot_product"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DotProduct"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre class="synopsis">ExpandMatrix
(M)</pre><p>
+ Expands a matrix just like we do on unquoted matrix input.
+ That is we expand any internal matrices as blocks. This is
+ a way to construct matrices out of smaller ones and this is
+ normally done automatically on input unless the matrix is quoted.
+ </p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre
class="synopsis">HermitianProduct (u,v)</pre><p>Aliases: <code class="function">InnerProduct</code></p><p>Get
the Hermitian product of two vectors. The vectors must be of the same size. This is a sesquilinear form
using the identity matrix.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sesquilinear_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/HermitianInnerProduct.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-I"></a>I</span></dt><dd><pre
class="synopsis">I (n)</pre><p>Aliases: <code class="function">eye</code></p><p>Return an identity matrix of
a given size, that is <code class="varname">n</code> by <code class="varname">n</code>. If <code
class="varname">n</code> is zero, returns <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Identity_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vec,msize)</pre><p>Return the index complement of a vector of indexes. Everything is one based. For
example for vector <strong class="userinput"><code>[2,3]</code></strong> and size
+<strong class="userinput"><code>5</code></strong>, we return <strong
class="userinput"><code>[1,4,5]</code></strong>. If
+<code class="varname">msize</code> is 0, we always return <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Is
a matrix diagonal.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Check if a matrix is the identity matrix. Automatically returns <code
class="constant">false</code>
+ if the matrix is not square. Also works on numbers, in which
+ case it is equivalent to <strong class="userinput"><code>x==1</code></strong>. When <code
class="varname">x</code> is
+ <code class="constant">null</code> (we could think of that as a 0 by 0 matrix),
+ no error is generated and <code class="constant">false</code> is returned.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Is a matrix lower triangular. That is, are all the entries
above the diagonal zero.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Check if a matrix is non-negative, that is if each element
is non-negative.
+ Do not confuse positive matrices with positive semi-definite matrices.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Check if a matrix is positive, that is if each element is
+positive (and hence real). In particular, no element is 0. Do not confuse
+positive matrices with positive definite matrices.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Check if a matrix is a matrix of rational (non-complex)
+numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Check if a matrix is a matrix of real (non-complex) numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>
+ Check if a matrix is square, that is its width is equal to
+ its height.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Is a matrix upper triangular? That is, a matrix is upper
triangular if all the entries below the diagonal are zero.</p></dd><dt><span class="term"><a
name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Check if a matrix is a matrix of numbers only. Many internal
+functions make this check. Values can be any number including complex numbers.</p></dd><dt><span
class="term"><a name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector
(v)</pre><p>
+ Is argument a horizontal or a vertical vector. Genius does
+ not distinguish between a matrix and a vector and a vector
+ is just a 1 by <code class="varname">n</code> or <code class="varname">n</code> by 1 matrix.
+ </p></dd><dt><span class="term"><a name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre
class="synopsis">IsZero (x)</pre><p>Check if a matrix is composed of all zeros. Also works on numbers, in
which
+ case it is equivalent to <strong class="userinput"><code>x==0</code></strong>. When <code
class="varname">x</code> is
+ <code class="constant">null</code> (we could think of that as a 0 by 0 matrix),
+ no error is generated and <code class="constant">true</code> is returned as the condition is
+ vacuous.
+ </p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Returns a copy of the matrix <code class="varname">M</code> with all the entries above the
diagonal set to zero.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Aliases: <code class="function">diag</code></p><p>Make diagonal matrix from a vector.
Alternatively you can pass
+ in the values to put on the diagonal as arguments. So
+ <strong class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> is the same as
+ <strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Make column vector out of matrix by putting columns above
+ each other. Returns <code class="constant">null</code> when given <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>
+ Calculate the product of all elements in a matrix or vector.
+ That is we multiply all the elements and return a number that
+ is the product of all the elements.
+ </p></dd><dt><span class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre
class="synopsis">MatrixSum (A)</pre><p>
+ Calculate the sum of all elements in a matrix or vector. That is
+ we add all the elements and return a number that is the
+ sum of all the elements.
+ </p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Calculate the sum of squares of all elements in a matrix
+ or vector.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Returns a row vector of the indices of nonzero columns in the matrix <code
class="varname">M</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Returns a row vector of the indices of nonzero elements in the vector <code
class="varname">v</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Get the outer product of two vectors. That is, suppose that
+<code class="varname">u</code> and <code class="varname">v</code> are vertical vectors, then
+the outer product is <strong class="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span
class="term"><a name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre
class="synopsis">ReverseVector (v)</pre><p>Reverse elements in a vector. Return <code
class="constant">null</code> if given <code class="constant">null</code></p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Calculate sum
of each row in a matrix and return a vertical
+vector with the result.</p></dd><dt><span class="term"><a
name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre class="synopsis">RowSumSquares
(m)</pre><p>Calculate sum of squares of each row in a matrix and return a vertical vector with the
results.</p></dd><dt><span class="term"><a name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre
class="synopsis">RowsOf (M)</pre><p>Gets the rows of a matrix as a vertical vector. Each element
+of the vector is a horizontal vector that is the corresponding row of
+<code class="varname">M</code>. This function is useful if you wish to loop over the
+rows of a matrix. For example, as <strong class="userinput"><code>for r in RowsOf(M) do
+something(r)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-SetMatrixSize"></a>SetMatrixSize</span></dt><dd><pre class="synopsis">SetMatrixSize
(M,rows,columns)</pre><p>Make new matrix of given size from old one. That is, a new
+ matrix will be returned to which the old one is copied. Entries that
+ don't fit are clipped and extra space is filled with zeros.
+ If <code class="varname">rows</code> or <code class="varname">columns</code> are zero
+ then <code class="constant">null</code> is returned.
+ </p></dd><dt><span class="term"><a
name="gel-function-ShuffleVector"></a>ShuffleVector</span></dt><dd><pre class="synopsis">ShuffleVector
(v)</pre><p>Shuffle elements in a vector. Return <code class="constant">null</code> if given <code
class="constant">null</code>.</p><p>Version 1.0.13 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SortVector"></a>SortVector</span></dt><dd><pre class="synopsis">SortVector
(v)</pre><p>Sort vector elements in an increasing order.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroColumns"></a>StripZeroColumns</span></dt><dd><pre
class="synopsis">StripZeroColumns (M)</pre><p>Removes any all-zero columns of <code
class="varname">M</code>.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroRows"></a>StripZeroRows</span></dt><dd><pre class="synopsis">StripZeroRows
(M)</pre><p>Removes any all-zero rows of <code class="varname">M</code>.</p></dd><dt><span class="term"><a
name="gel-function-Submatrix"></a>Sub
matrix</span></dt><dd><pre class="synopsis">Submatrix (m,r,c)</pre><p>Return column(s) and row(s) from a
matrix. This is
+just equivalent to <strong class="userinput"><code>m@(r,c)</code></strong>. <code class="varname">r</code>
+and <code class="varname">c</code> should be vectors of rows and columns (or single numbers if only one row
or column is needed).</p></dd><dt><span class="term"><a
name="gel-function-SwapRows"></a>SwapRows</span></dt><dd><pre class="synopsis">SwapRows
(m,row1,row2)</pre><p>Swap two rows in a matrix.</p></dd><dt><span class="term"><a
name="gel-function-UpperTriangular"></a>UpperTriangular</span></dt><dd><pre class="synopsis">UpperTriangular
(M)</pre><p>Returns a copy of the matrix <code class="varname">M</code> with all the entries below the
diagonal set to zero.</p></dd><dt><span class="term"><a
name="gel-function-columns"></a>columns</span></dt><dd><pre class="synopsis">columns (M)</pre><p>Get the
number of columns of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-elements"></a>elements</span></dt><dd><pre class="synopsis">elements (M)</pre><p>Get the
total number of elements of a matrix. This is the
+number of columns times the number of rows.</p></dd><dt><span class="term"><a
name="gel-function-ones"></a>ones</span></dt><dd><pre class="synopsis">ones (rows,columns...)</pre><p>Make an
matrix of all ones (or a row vector if only one argument is given). Returns <code
class="constant">null</code> if either rows or columns are zero.</p></dd><dt><span class="term"><a
name="gel-function-rows"></a>rows</span></dt><dd><pre class="synopsis">rows (M)</pre><p>Get the number of
rows of a matrix.</p></dd><dt><span class="term"><a name="gel-function-zeros"></a>zeros</span></dt><dd><pre
class="synopsis">zeros (rows,columns...)</pre><p>Make a matrix of all zeros (or a row vector if only one
argument is given). Returns <code class="constant">null</code> if either rows or columns are
zero.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2 class="title" style="cl
ear: both"><a name="genius-gel-function-list-linear-algebra"></a>Linear Algebra</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Get the auxiliary unit matrix of size <code
class="varname">n</code>. This is a square matrix with that is all zero except the
+superdiagonal being all ones. It is the Jordan block matrix of one zero eigenvalue.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a>
for more information on Jordan Canonical Form.
+ </p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm
(v,A,w)</pre><p>Evaluate (v,w) with respect to the bilinear form given by the matrix A.</p></dd><dt><span
class="term"><a name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Return a function that evaluates two vectors with respect
to the bilinear form given by A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Aliases: <code
class="function">CharPoly</code></p><p>Get the characteristic polynomial as a vector. That is, return
+the coefficients of the polynomial starting with the constant term. This is
+the polynomial defined by <strong class="userinput"><code>det(M-xI)</code></strong>. The roots of this
+polynomial are the eigenvalues of <code class="varname">M</code>.
+See also <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">CharacteristicPolynomialFunction</a>.
+</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Get the characteristic polynomial as a
function. This is
+the polynomial defined by <strong class="userinput"><code>det(M-xI)</code></strong>. The roots of this
+polynomial are the eigenvalues of <code class="varname">M</code>.
+See also <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.
+</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre class="synopsis">ColumnSpace
(M)</pre><p>Get a basis matrix for the columnspace of a matrix. That is,
+return a matrix whose columns are the basis for the column space of
+<code class="varname">M</code>. That is the space spanned by the columns of
+<code class="varname">M</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_and_column_spaces";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre
class="synopsis">CommutationMatrix (m, n)</pre><p>Return the commutation matrix <strong
class="userinput"><code>K(m,n)</code></strong>, which is the unique <strong
class="userinput"><code>m*n</code></strong> by
+ <strong class="userinput"><code>m*n</code></strong> matrix such that <strong
class="userinput"><code>K(m,n) * MakeVector(A) = MakeVector(A.')</code></strong> for all <code
class="varname">m</code> by <code class="varname">n</code>
+ matrices <code class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre class="synopsis">CompanionMatrix
(p)</pre><p>Companion matrix of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Conjugate transpose of a matrix (adjoint). This is the
+ same as the <strong class="userinput"><code>'</code></strong> operator.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Conjugate_transpose";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ConjugateTranspose"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Aliases: <code class="function">convol</code></p><p>Calculate convolution of two horizontal
vectors.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Calculate convolution of two horizontal vectors. Return
+result as a vector and not added together.</p></dd><dt><span class="term"><a
name="gel-function-CrossProduct"></a>CrossProduct</span></dt><dd><pre class="synopsis">CrossProduct
(v,w)</pre><p>CrossProduct of two vectors in R<sup>3</sup> as
+ a column vector.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cross_product"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre
class="synopsis">DeterminantalDivisorsInteger (M)</pre><p>Get the determinantal divisors of an integer
matrix.</p></dd><dt><span class="term"><a name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre
class="synopsis">DirectSum (M,N...)</pre><p>Direct sum of matrices.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DirectSumMatrixVector"></a>DirectSumMatrixVector</span></dt><dd><pre
class="synopsis">DirectSumMatrixVector (v)</pre><p>Direct sum of a vector of matrices.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Aliases: <code class="function">eig</code></p><p>Get the eigenvalues of a square matrix.
+ Currently only works for matrices of size up to 4 by 4, or for
+ triangular matrices (for which the eigenvalues are on the
+ diagonal).
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Get the eigenvectors of a square matrix. Optionally get also
+the eigenvalues and their algebraic multiplicities.
+ Currently only works for matrices of size up to 2 by 2.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Apply the Gram-Schmidt process (to the columns) with respect to
+inner product given by <code class="varname">B</code>. If <code class="varname">B</code> is not
+given then the standard Hermitian product is used. <code class="varname">B</code> can
+either be a sesquilinear function of two arguments or it can be a matrix giving
+a sesquilinear form. The vectors will be made orthonormal with respect to
+<code class="varname">B</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/GramSchmidtOrthogonalization";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span></dt><dd><pre class="synopsis">HankelMatrix
(c,r)</pre><p>Hankel matrix, a matrix whose skew-diagonals are constant. <code class="varname">c</code> is
the first row and <code class="varname">r</code> is the
+ last column. It is assumed that both arguments are vectors and the last element of <code
class="varname">c</code> is the same
+ as the first element of <code class="varname">r</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hankel_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Hilbert matrix of order <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-Image"></a>Image</span></dt><dd><pre
class="synopsis">Image (T)</pre><p>Get the image (columnspace) of a linear transform.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_and_column_spaces";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre
class="synopsis">InfNorm (v)</pre><p>Get the Inf Norm of a vector, sometimes called the sup norm or the max
norm.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Get the invariant factors of a square integer
matrix.</p></dd><dt><span class="term"><a
name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatrix</span></dt><dd><pre
class="synopsis">InverseHilbertMatrix (n)</pre><p>Inverse Hilbert matrix of order <code
class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Is a matrix Hermitian. That is, is it equal to its conjugate transpose.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hermitian_matrix";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/HermitianMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsInSubspace"></a>IsInSubspace</span></dt><dd><pre class="synopsis">IsInSubspace
(v,W)</pre><p>Test if a vector is in a subspace.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Is a matrix (or number) invertible (Integer matrix is invertible if and only if it is invertible
over the integers).</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Is a matrix (or number) invertible over a
field.</p></dd><dt><span class="term"><a name="gel-function-IsNormal"></a>IsNormal</span></dt><dd><pre
class="synopsis">IsNormal (M)</pre><p>Is <code class="varname">M</code> a normal matrix. That is,
+ does <strong class="userinput"><code>M*M' == M'*M</code></strong>.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/NormalMatrix"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalMatrix.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Is <code class="varname">M</code> a Hermitian positive
definite matrix. That is if
+<strong class="userinput"><code>HermitianProduct(M*v,v)</code></strong> is always strictly positive for
+any vector <code class="varname">v</code>.
+<code class="varname">M</code> must be square and Hermitian to be positive definite.
+The check that is performed is that every principal submatrix has a non-negative
+determinant.
+(See <a class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>
+ Note that some authors (for example Mathworld) do not require that
+ <code class="varname">M</code> be Hermitian, and then the condition is
+ on the real part of the inner product, but we do not take this
+ view. If you wish to perform this check, just check the
+ Hermitian part of the matrix <code class="varname">M</code> as follows:
+ <strong class="userinput"><code>IsPositiveDefinite(M+M')</code></strong>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive-definite_matrix";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Is <code class="varname">M</code> a Hermitian positive
semidefinite matrix. That is if
+<strong class="userinput"><code>HermitianProduct(M*v,v)</code></strong> is always non-negative for
+any vector <code class="varname">v</code>.
+<code class="varname">M</code> must be square and Hermitian to be positive semidefinite.
+The check that is performed is that every principal submatrix has a non-negative
+determinant.
+(See <a class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>
+ Note that some authors do not require that
+ <code class="varname">M</code> be Hermitian, and then the condition is
+ on the real part of the inner product, but we do not take this
+ view. If you wish to perform this check, just check the
+ Hermitian part of the matrix <code class="varname">M</code> as follows:
+ <strong class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian
(M)</pre><p>Is a matrix skew-Hermitian. That is, is the conjugate transpose equal to negative of the
matrix.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/SkewHermitianMatrix"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre
class="synopsis">IsUnitary (M)</pre><p>Is a matrix unitary? That is, does
+ <strong class="userinput"><code>M'*M</code></strong> and <strong
class="userinput"><code>M*M'</code></strong>
+ equal the identity.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/UnitaryTransformation"; target="_top">Planetmath</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/UnitaryMatrix.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-JordanBlock"></a>JordanBlock</span></dt><dd><pre class="synopsis">JordanBlock
(n,lambda)</pre><p>Aliases: <code class="function">J</code></p><p>Get the Jordan block corresponding to the
eigenvalue
+ <code class="varname">lambda</code> with multiplicity <code class="varname">n</code>.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre
class="synopsis">Kernel (T)</pre><p>Get the kernel (nullspace) of a linear transform.</p><p>
+ (See <a class="link" href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)
+ </p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Aliases: <code class="function">TensorProduct</code></p><p>
+ Compute the Kronecker product (tensor product in standard basis)
+ of two matrices.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
+ Get the LU decomposition of <code class="varname">A</code>, that is
+ find a lower triangular matrix and upper triangular
+ matrix whose product is <code class="varname">A</code>.
+ Store the result in the <code class="varname">L</code> and
+ <code class="varname">U</code>, which should be references. It returns <code
class="constant">true</code>
+ if successful.
+ For example suppose that A is a square matrix, then after running:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LUDecomposition(A,&L,&U)</code></strong>
+</pre><p>
+ You will have the lower matrix stored in a variable called
+ <code class="varname">L</code> and the upper matrix in a variable called
+ <code class="varname">U</code>.
+ </p><p>
+ This is the LU decomposition of a matrix aka Crout and/or Cholesky
+ reduction.
+ (ISBN 0-201-11577-8 pp.99-103)
+ The upper triangular matrix features a diagonal
+ of values 1 (one). This is not Doolittle's Method, which features
+ the 1's diagonal on the lower matrix.
+ </p><p>
+ Not all matrices have LU decompositions, for example
+ <strong class="userinput"><code>[0,1;1,0]</code></strong> does not and this function returns
+ <code class="constant">false</code> in this case and sets <code class="varname">L</code>
+ and <code class="varname">U</code> to <code class="constant">null</code>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Get the <code class="varname">i</code>-<code class="varname">j</code>
minor of a matrix.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Minor"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Return the columns that are not the pivot columns of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm (v,p...)</pre><p>Aliases: <code
class="function">norm</code></p><p>Get the p Norm (or 2 Norm if no p is supplied) of a
vector.</p></dd><dt><span class="term"><a name="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre
class="synopsis">NullSpace (T)</pre><p>Get the nullspace of a matrix. That is the kernel of the
+ linear mapping that the matrix represents. This is returned
+ as a matrix whose column space is the nullspace of
+ <code class="varname">T</code>.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Nullspace"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre
class="synopsis">Nullity (M)</pre><p>Aliases: <code class="function">nullity</code></p><p>Get the nullity of
a matrix. That is, return the dimension of
+the nullspace; the dimension of the kernel of <code class="varname">M</code>.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Nullity"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre><p>Get the orthogonal complement of the
columnspace.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Return pivot columns of a matrix, that is columns that have a leading 1 in row reduced form.
Also returns the row where they occur.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Projection of vector <code class="varname">v</code> onto subspace
+<code class="varname">W</code> with respect to inner product given by
+<code class="varname">B</code>. If <code class="varname">B</code> is not given then the standard
+Hermitian product is used. <code class="varname">B</code> can either be a sesquilinear
+function of two arguments or it can be a matrix giving a sesquilinear form.
+ </p></dd><dt><span class="term"><a
name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre class="synopsis">QRDecomposition
(A, Q)</pre><p>
+ Get the QR decomposition of a square matrix <code class="varname">A</code>,
+ returns the upper triangular matrix <code class="varname">R</code>
+ and sets <code class="varname">Q</code> to the orthogonal (unitary) matrix.
+ <code class="varname">Q</code> should be a reference or <code class="constant">null</code> if you
don't
+ want any return.
+ For example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>R
= QRDecomposition(A,&Q)</code></strong>
+</pre><p>
+ You will have the upper triangular matrix stored in
+ a variable called
+ <code class="varname">R</code> and the orthogonal (unitary) matrix stored in
+ <code class="varname">Q</code>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Return the Rayleigh quotient (also called the Rayleigh-Ritz
quotient or ratio) of a matrix and a vector.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)</pre><p>Find eigenvalues of <code
class="varname">A</code> using the Rayleigh
+ quotient iteration method. <code class="varname">x</code> is a guess
+ at a eigenvector and could be random. It should have
+ nonzero imaginary part if it will have any chance at finding
+ complex eigenvalues. The code will run at most
+ <code class="varname">maxiter</code> iterations and return <code class="constant">null</code>
+ if we cannot get within an error of <code class="varname">epsilon</code>.
+ <code class="varname">vecref</code> should either be <code class="constant">null</code> or a
reference
+ to a variable where the eigenvector should be stored.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> for
more information on Rayleigh quotient.
+ </p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span></dt><dd><pre
class="synopsis">Rank (M)</pre><p>Aliases: <code class="function">rank</code></p><p>Get the rank of a
matrix.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</p></dd><dt><span
class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Return the matrix corresponding
to rotation around origin in R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
x-axis.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (angle)</pre><p>Return the matrix corresponding to rotation around origin in
R<sup>3</sup> about the
y-axis.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
z-axis.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Get a basis matrix for the rowspace of a matrix.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Evaluate (v,w) with respect to the sesquilinear form given
by the matrix A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Return a function that evaluates two vectors with
respect to the sesquilinear form given by A.</p></dd><dt><span class="term"><a name="gel
-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returns the Smith normal form of a matrix over fields (will
end up with 1's on the diagonal).</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Solve linear system Mx=V, return solution V if there
is a unique solution, <code class="constant">null</code> otherwise. Extra two reference parameters can
optionally be used to get the reduced M and V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Return the Toeplitz matrix constructed given the first column c
+and (optionally) the first row r. If only the column c is given then it is
+conjugated and the nonconjugated version is used for the first row to give a
+Hermitian matrix (if the first element is real of course).</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Aliases: <code class="function">trace</code></p><p>Calculate the trace of
a matrix. That is the sum of the diagonal elements.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Transpose of a matrix. This is the same as the
+ <strong class="userinput"><code>.'</code></strong> operator.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Aliases: <code class="function">vander</code></p><p>Return the
Vandermonde matrix.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>The angle of two vectors with respect to inner product given by
+<code class="varname">B</code>. If <code class="varname">B</code> is not given then the standard
+Hermitian product is used. <code class="varname">B</code> can either be a sesquilinear
+function of two arguments or it can be a matrix giving a sesquilinear form.
+</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>The direct sum of the vector spaces M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersection of the subspaces given by M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>The sum of the vector spaces M and N, that is {w | w=m+n, m
in M, n in N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Aliases: <code class="function">Adjugate</code></p><p>Get the classical
adjoint (adjugate) of a matrix.</p></dd><dt><span class="term"><a name="gel-function-cref"></a>cref</spa
n></dt><dd><pre class="synopsis">cref (M)</pre><p>Aliases: <code class="function">CREF</code> <code
class="function">ColumnReducedEchelonForm</code></p><p>Compute the Column Reduced Echelon
Form.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Aliases: <code class="function">Determinant</code></p><p>Get the determinant
of a matrix.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Aliases: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Get the row echelon form of a matrix. That is, apply gaussian
+elimination but not backaddition to <code class="varname">M</code>. The pivot rows are
+divided to make all pivots 1.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Aliases: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Get the reduced row echelon form of a matrix. That is,
apply gaussian elimination together with backaddition to <code class="varname">M</code>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2 class="title" style="clear: both"><
a name="genius-gel-function-list-combinatorics"></a>Combinatorics</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Get <code
class="varname">n</code>th Catalan number.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Get all combinations of k numbers from 1 to n as a vector of vectors.
+ (See also <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)
+</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Factorial: <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre
class="synopsis">FallingFactorial (n,k)</pre><p>Falling factorial: <strong class="userinput"><code>(n)_k =
n(n-1)...(n-(k-1))</code></strong></p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre
class="synopsis">Fibonacci (x)</pre><p>Aliases: <code class="function">fib</code></p><p>
+ Calculate <code class="varname">n</code>th Fibonacci number. That
+ is the number defined recursively by
+ <strong class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</code></strong>
+ and
+ <strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
+ Calculate the Frobenius number. That is calculate smallest
+ number that cannot be given as a non-negative integer linear
+ combination of a given vector of non-negative integers.
+ The vector can be given as separate numbers or a single vector.
+ All the numbers given should have GCD of 1.
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(combining_rule)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>
+ Find the vector <code class="varname">c</code> of non-negative integers
+ such that taking the dot product with <code class="varname">v</code> is
+ equal to n. If not possible returns <code class="constant">null</code>. <code
class="varname">v</code>
+ should be given sorted in increasing order and should consist
+ of non-negative integers.
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coeffi
cients. Takes a vector of
+ <code class="varname">k</code>
+ non-negative integers and computes the multinomial coefficient.
+ This corresponds to the coefficient in the homogeneous polynomial
+ in <code class="varname">k</code> variables with the corresponding powers.
+ </p><p>
+ The formula for <strong class="userinput"><code>Multinomial(a,b,c)</code></strong>
+ can be written as:
+</p><pre class="programlisting">(a+b+c)! / (a!b!c!)
+</pre><p>
+ In other words, if we would have only two elements, then
+<strong class="userinput"><code>Multinomial(a,b)</code></strong> is the same thing as
+<strong class="userinput"><code>Binomial(a+b,a)</code></strong> or
+<strong class="userinput"><code>Binomial(a+b,b)</code></strong>.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Get combination that would come after v in call to
+combinations, first combination should be <strong class="userinput"><code>[1:k]</code></strong>. This
+function is useful if you have many combinations to go through and you don't
+want to waste memory to store them all.
+ </p><p>
+ For example with Combinations you would normally write a loop like:
+ </p><pre class="screen"><strong class="userinput"><code>for n in Combinations (4,6) do (
+ SomeFunction (n)
+);</code></strong>
+</pre><p>
+ But with NextCombination you would write something like:
+ </p><pre class="screen"><strong class="userinput"><code>n:=[1:4];
+do (
+ SomeFunction (n)
+) while not IsNull(n:=NextCombination(n,6));</code></strong>
+</pre><p>
+ See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Get the Pascal's triangle as a matrix. This will return
+ an <code class="varname">i</code>+1 by <code class="varname">i</code>+1 lower diagonal
+ matrix that is the Pascal's triangle after <code class="varname">i</code>
+ iterations.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Get all permutations of <code class="varname">k</code> numbers from 1 to <code
class="varname">n</code> as a vector of vectors.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Aliases: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k =
n(n+1)...(n+(k-1)).</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Aliases: <code
class="function">StirlingS1</code></p><p>Stirling number of the first kind.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/StirlingNumbersOfTheFirstKind";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Aliases: <code
class="function">StirlingS2</code></p><p>Stirling number of the second kind.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/StirlingNumbersSecondKind";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangular
(nth)</pre><p>Calculate the <code class="varname">n</code>th triangular number.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/TriangularNumbers"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-nCr"></a>nCr</span></dt><dd><pre
class="synopsis">nCr (n,r)</pre><p>Aliases: <code class="function">Binomial</code></p><p>Calculate
combinations, that is, the binomial coefficient.
+ <code class="varname">n</code> can be any real number.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre
class="synopsis">nPr (n,r)</pre><p>Calculate the number of permutations of size
+ <code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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diff --git a/help/C/ch11s11.html b/help/C/ch11s11.html
new file mode 100644
index 0000000..d38c659
--- /dev/null
+++ b/help/C/ch11s11.html
@@ -0,0 +1,120 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Calculus</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
href="ch11s10.html" title="Combinatorics"><link rel="next" href="ch11s12.html" title="Functions"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Calculus</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s10.html">Prev</a> </td><th width="60%"
align="center">Chapter 11. List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s12.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="ge
nius-gel-function-list-calculus"></a>Calculus</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integration of f by Composite Simpson's Rule on the
interval [a,b] with n subintervals with error of max(f'''')*h^4*(b-a)/180, note that n should be even.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)</pre><p>Integration
of f by Composite Simpson's Rule on the interval [a,b] with the number of steps calculated by the fourth
derivative bound and the desired tolerance.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Attempt to calculate derivative by trying first symbolically and then numerically.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Return a function that is the even periodic extension of
+<code class="function">f</code> with half period <code class="varname">L</code>. That
+is a function defined on the interval <strong class="userinput"><code>[0,L]</code></strong>
+extended to be even on <strong class="userinput"><code>[-L,L]</code></strong> and then
+extended to be periodic with period <strong class="userinput"><code>2*L</code></strong>.</p><p>
+ See also
+ <a class="link" href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a>
+ and
+ <a class="link" href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Return a function that is a Fourier series with the
+coefficients given by the vectors <code class="varname">a</code> (sines) and
+<code class="varname">b</code> (cosines). Note that <strong class="userinput"><code>a@(1)</code></strong> is
+the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
+the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, while
+<strong class="userinput"><code>b@(n)</code></strong> refers to the term
+<strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Either <code class="varname">a</code>
+or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Try to calculate an infinite product for a single parameter
function.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Try to calculate an infinite product for a
double parameter function with func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Try to calculate an infinite sum for a single parameter function.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,inc)</pre><p>Try to calculate an infinite sum for a double
parameter function with func(arg,n).</p></dd><d
t><span class="term"><a name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre
class="synopsis">IsContinuous (f,x0)</pre><p>Try and see if a real-valued function is continuous at x0 by
calculating the limit there.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Test for differentiability by approximating the left and
right limits and comparing.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calculate the left limit of a real-valued function at x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Calculate the
limit of a real-valued function at x0. Tries to calculate both left and right limits.</p></dd><dt><span
class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</sp
an></dt><dd><pre class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integration by midpoint
rule.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Aliases: <code
class="function">NDerivative</code></p><p>Attempt to calculate numerical derivative.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Return a vector of vectors <strong
class="userinput"><code>[a,b]</code></strong>
+where <code class="varname">a</code> are the cosine coefficients and
+<code class="varname">b</code> are the sine coefficients of
+the Fourier series of
+<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
+on <strong class="userinput"><code>[-L,L]</code></strong> and extended periodically) with coefficients
+up to <code class="varname">N</code>th harmonic computed numerically. The coefficients are
+computed by numerical integration using
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
+<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
+on <strong class="userinput"><code>[-L,L]</code></strong> and extended periodically) with coefficients
+up to <code class="varname">N</code>th harmonic computed numerically. This is the
+trigonometric real series composed of sines and cosines. The coefficients are
+computed by numerical integration using
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
+the cosine Fourier series of
+<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
+we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
+take the even periodic extension and compute the Fourier series, which
+only has cosine terms. The series is computed up to the
+<code class="varname">N</code>th harmonic. The coefficients are
+computed by numerical integration using
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.
+Note that <strong class="userinput"><code>a@(1)</code></strong> is
+the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
+the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
+<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
+we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
+take the even periodic extension and compute the Fourier series, which
+only has cosine terms. The series is computed up to the
+<code class="varname">N</code>th harmonic. The coefficients are
+computed by numerical integration using
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
+the sine Fourier series of
+<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
+we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
+take the odd periodic extension and compute the Fourier series, which
+only has sine terms. The series is computed up to the
+<code class="varname">N</code>th harmonic. The coefficients are
+computed by numerical integration using
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
+<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
+we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
+take the odd periodic extension and compute the Fourier series, which
+only has sine terms. The series is computed up to the
+<code class="varname">N</code>th harmonic. The coefficients are
+computed by numerical integration using
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration by rule set in NumericalIntegralFunction of f
from a to b using NumericalIntegralSteps steps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Attempt to calculate numerical left
derivative.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Attempt to
calculate the limit of f(step_fun(i)) as i goes from 1 to N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">Nume
ricalRightDerivative (f,x0)</pre><p>Attempt to calculate numerical right derivative.</p></dd><dt><span
class="term"><a name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
+<code class="function">f</code> with half period <code class="varname">L</code>. That
+is a function defined on the interval <strong class="userinput"><code>[0,L]</code></strong>
+extended to be odd on <strong class="userinput"><code>[-L,L]</code></strong> and then
+extended to be periodic with period <strong class="userinput"><code>2*L</code></strong>.</p><p>
+ See also
+ <a class="link" href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>
+ and
+ <a class="link" href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Compute one-sided derivative using five point
formula.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Compute one-sided derivative using three-point
formula.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Return a function that is the periodic extension of
+<code class="function">f</code> defined on the interval <strong class="userinput"><code>[a,b]</code></strong>
+and has period <strong class="userinput"><code>b-a</code></strong>.</p><p>
+ See also
+ <a class="link" href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a>
+ and
+ <a class="link" href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre class="synopsis">RightLimit
(f,x0)</pre><p>Calculate the right limit of a real-valued function at x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Compute two-sided derivative using five-point
formula.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Compute two-sided derivative using three-point
formula.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Functions</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
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Solving"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
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align="center">Functions</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s13.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name
="genius-gel-function-list-functions"></a>Functions</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Aliases: <code class="function">Arg</code> <code
class="function">arg</code></p><p>argument (angle) of complex number.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0 (x)</pre><p>Bessel
function of the first kind of order 0. Only implemented for real numbers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Bessel
function of the first kind of order 1. Only implemented for real numbers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Bessel
function of the first kind of order <code class="varname">n</code>. Only implemented for real numbers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Bessel
function of the second kind of order 0. Only implemented for real numbers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Bessel
function of the second kind of order 1. Only implemented for real numbers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Bessel
function of the second kind of order <code class="varname">n</code>. Only implemented for real
numbers.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Aliases: <code class="function">erf</code></p><p>The error function, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/ErrorFunction"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FejerKernel"></a>FejerKernel</span></dt><dd><pre class="synopsis">FejerKernel
(n,t)</pre><p>Fejer kernel of order <code class="varname">n</code> evaluated at
+ <code class="varname">t</code></p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/FejerKernel"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Aliases: <code class="function">Gamma</code></p><p>The Gamma function. Currently only
implemented for real values.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returns 1 if and only if all elements are equal.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>
+ The principal branch of Lambert W function computed for only
+ real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>.
+ That is, <code class="function">LambertW</code> is the inverse of
+ the expression <strong class="userinput"><code>x*e^x</code></strong>. Even for
+ real <code class="varname">x</code> this expression is not one to one and
+ therefore has two branches over <strong class="userinput"><code>[-1/e,0)</code></strong>.
+ See <a class="link" href="ch11s12.html#gel-function-LambertWm1"><code
class="function">LambertWm1</code></a> for the other real branch.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>
+ The minus-one branch of Lambert W function computed for only
+ real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>
+ and less than 0.
+ That is, <code class="function">LambertWm1</code> is the second
+ branch of the inverse of <strong class="userinput"><code>x*e^x</code></strong>.
+ See <a class="link" href="ch11s12.html#gel-function-LambertW"><code
class="function">LambertW</code></a> for the principal branch.
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Find the first value where f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Moebius mapping of the disk to itself mapping a to 0.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity
respectively.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Moebius mapping using the cross ratio taking
infinity to infinity and z2,z3 to 1 and 0 respectively.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 1 and z3,z4 to 0 and infinity respectively.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 0 and z2,z4 to 1 and infinity respectively.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poisson kernel on D(0,R) (not normalized to
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Aliases: <code class="function">zeta</code></p><p>The Riemann zeta
function. Currently only implemented for real values.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>The unit step function is 0 for x<0, 1 otherwise. This is the
integral of the Dirac Delta function. Also called the Heaviside function.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>
+ The <code class="function">cis</code> function, that is the same as
+ <strong class="userinput"><code>cos(x)+1i*sin(x)</code></strong>
+ </p></dd><dt><span class="term"><a name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre
class="synopsis">deg2rad (x)</pre><p>Convert degrees to radians.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Convert
radians to degrees.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Calculates the unnormalized sinc function, that is
+ <strong class="userinput"><code>sin(x)/x</code></strong>.
+ If you want the normalized function call <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ </p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
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class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both
"><a name="genius-gel-function-list-equation-solving"></a>Equation Solving</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre class="synopsis">CubicFormula
(p)</pre><p>
+ Compute roots of a cubic (degree 3) polynomial using the
+ cubic formula. The polynomial should be given as a
+ vector of coefficients. That is
+ <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> corresponds to the vector
+ <strong class="userinput"><code>[1,2,0,4]</code></strong>.
+ Returns a column vector of the three solutions. The first solution is always
+ the real one as a cubic always has one real solution.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
+ Use classical Euler's method to numerically solve y'=f(x,y) for
+ initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
+ <code class="varname">x1</code> with <code class="varname">n</code> increments,
+ returns <code class="varname">y</code> at <code class="varname">x1</code>.
+ Unless you explicitly want to use Euler's method, you should really
+ think about using
+ <a class="link" href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a>
+ for solving ODE.
+ </p><p>
+ Systems can be solved by just having <code class="varname">y</code> be a
+ (column) vector everywhere. That is, <code class="varname">y0</code> can
+ be a vector in which case <code class="varname">f</code> should take a number
+ <code class="varname">x</code> and a vector of the same size for the second
+ argument and should return a vector of the same size.
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
+ Use classical Euler's method to numerically solve y'=f(x,y) for
+ initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
+ <code class="varname">x1</code> with <code class="varname">n</code> increments,
+ returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ <code class="varname">x</code> and <code class="varname">y</code> values.
+ Unless you explicitly want to use Euler's method, you should really
+ think about using
+ <a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a>
+ for solving ODE.
+ Suitable
+ for plugging into
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>line =
EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential
growth");</code></strong>
+</pre><p>
+ </p><p>
+ Systems can be solved by just having <code class="varname">y</code> be a
+ (column) vector everywhere. That is, <code class="varname">y0</code> can
+ be a vector in which case <code class="varname">f</code> should take a number
+ <code class="varname">x</code> and a vector of the same size for the second
+ argument and should return a vector of the same size.
+ </p><p>
+ The output for a system is still a n by 2 matrix with the second
+ entry being a vector. If you wish to plot the line, make sure to use row vectors, and then
flatten the matrix with
+ <a class="link" href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>,
+ and pick out the right columns. Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>lines =
EulersMethodFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,500);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>lines =
ExpandMatrix(lines);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>firstline =
lines@(,[1,2]);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>secondline =
lines@(,[1,3]);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","First");</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</code></strong>
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Find root of a function using the bisection method.
+ <code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
+ <strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
+ <code class="varname">TOL</code> is the desired tolerance and
+<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The
function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where
<code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is
the last value computed, and <code class="varname">iteration</code> is the number of iterations
done.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Find root of a function using the method of
false position.
+ <code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
+ <strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
+ <code class="varname">TOL</code> is the desired tolerance and
+<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The
function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where
<code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is
the last value computed, and <code class="varname">iteration</code> is the number of iterations
done.</p></dd><dt><span class="term"><a
name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Find root of a function using the Muller's
method.
+ <code class="varname">TOL</code> is the desired tolerance and
+<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The
function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where
<code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is
the last value computed, and <code class="varname">iteration</code> is the number of iterations
done.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Find root of a function using the secant method.
+ <code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
+ <strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
+ <code class="varname">TOL</code> is the desired tolerance and
+<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The
function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where
<code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is
the last value computed, and <code class="varname">iteration</code> is the number of iterations
done.</p></dd><dt><span class="term"><a
name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,guess,epsilon,maxn)</pre><p>Find zeros using Halley's method. <code class="varname">f</code> is
+ the function, <code class="varname">df</code> is the derivative of
+ <code class="varname">f</code>, and <code class="varname">ddf</code> is the second
derivative of
+ <code class="varname">f</code>. <code class="varname">guess</code> is the initial
+ guess. The function returns after two successive values are
+ within <code class="varname">epsilon</code> of each other, or after <code
class="varname">maxn</code> tries, in which case the function returns <code class="constant">null</code>
indicating failure.
+ </p><p>
+ See also <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a> and <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code class="function">SymbolicDerivative</code></a>.
+ </p><p>
+ Example to find the square root of 10:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Find zeros using Newton's method. <code class="varname">f</code> is
+ the function and <code class="varname">df</code> is the derivative of
+ <code class="varname">f</code>. <code class="varname">guess</code> is the initial
+ guess. The function returns after two successive values are
+ within <code class="varname">epsilon</code> of each other, or after <code
class="varname">maxn</code> tries, in which case the function returns <code class="constant">null</code>
indicating failure.
+ </p><p>
+ See also <a class="link" href="ch11s15.html#gel-function-NewtonsMethodPoly"><code
class="function">NewtonsMethodPoly</code></a> and <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code class="function">SymbolicDerivative</code></a>.
+ </p><p>
+ Example to find the square root of 10:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>
+ Compute roots of a polynomial (degrees 1 through 4)
+ using one of the formulas for such polynomials.
+ The polynomial should be given as a
+ vector of coefficients. That is
+ <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> corresponds to the vector
+ <strong class="userinput"><code>[1,2,0,4]</code></strong>.
+ Returns a column vector of the solutions.
+ </p><p>
+ The function calls
+ <a class="link" href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>,
+ <a class="link" href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a>, and
+ <a class="link" href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre
class="synopsis">QuadraticFormula (p)</pre><p>
+ Compute roots of a quadratic (degree 2) polynomial using the
+ quadratic formula. The polynomial should be given as a
+ vector of coefficients. That is
+ <strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> corresponds to the vector
+ <strong class="userinput"><code>[1,2,3]</code></strong>.
+ Returns a column vector of the two solutions.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>
+ Compute roots of a quartic (degree 4) polynomial using the
+ quartic formula. The polynomial should be given as a
+ vector of coefficients. That is
+ <strong class="userinput"><code>5*x^4 + 2*x + 1</code></strong> corresponds to the vector
+ <strong class="userinput"><code>[1,2,0,0,5]</code></strong>.
+ Returns a column vector of the four solutions.
+ </p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>
+ Use classical non-adaptive fourth order Runge-Kutta method to
+ numerically solve
+ y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
+ going to <code class="varname">x1</code> with <code class="varname">n</code>
+ increments, returns <code class="varname">y</code> at <code class="varname">x1</code>.
+ </p><p>
+ Systems can be solved by just having <code class="varname">y</code> be a
+ (column) vector everywhere. That is, <code class="varname">y0</code> can
+ be a vector in which case <code class="varname">f</code> should take a number
+ <code class="varname">x</code> and a vector of the same size for the second
+ argument and should return a vector of the same size.
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
+ Use classical non-adaptive fourth order Runge-Kutta method to
+ numerically solve
+ y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
+ going to <code class="varname">x1</code> with <code class="varname">n</code>
+ increments,
+ returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ <code class="varname">x</code> and <code class="varname">y</code> values. Suitable
+ for plugging into
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>line =
RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential
growth");</code></strong>
+</pre><p>
+ </p><p>
+ Systems can be solved by just having <code class="varname">y</code> be a
+ (column) vector everywhere. That is, <code class="varname">y0</code> can
+ be a vector in which case <code class="varname">f</code> should take a number
+ <code class="varname">x</code> and a vector of the same size for the second
+ argument and should return a vector of the same size.
+ </p><p>
+ The output for a system is still a n by 2 matrix with the second
+ entry being a vector. If you wish to plot the line, make sure to use row vectors, and then
flatten the matrix with
+ <a class="link" href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>,
+ and pick out the right columns. Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>lines =
RungeKuttaFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,100);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>lines =
ExpandMatrix(lines);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>firstline =
lines@(,[1,2]);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>secondline =
lines@(,[1,3]);</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","First");</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</code></strong>
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
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diff --git a/help/C/ch11s14.html b/help/C/ch11s14.html
new file mode 100644
index 0000000..00093d4
--- /dev/null
+++ b/help/C/ch11s14.html
@@ -0,0 +1,20 @@
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistics</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s13.html" title="Equation Solving"><link
rel="next" href="ch11s15.html" title="Polynomials"></head><body bgcolor="white" text="black" link="#0000FF"
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header"><tr><th colspan="3" align="center">Statistics</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
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class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Statistics</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average of an entire matrix.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral of the GaussFunction from 0 to <code
class="varname">x</code> (area under the normal curve).</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>The normalized Gauss distribution function (the normal curve).</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Aliases: <code class="function">median</code></p><p>Calculate median of
an entire matrix.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calculate median of each row in a matrix and return a column
+ vector of the medians.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdevp</code></p><p>Calculate the population standard deviations of rows of a matrix and
return a vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdev</code></p><p>Calculate the standard deviations of rows of a matrix and return a
vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Aliases: <code class="function">stdev</code></p><p>Calculate
the standard deviation of a whole matrix.</p></dd></dl></div></div><div class="navfooter"><hr><table widt
h="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Polynomials</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s14.html" title="Statistics"><link
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header"><tr><th colspan="3" align="center">Polynomials</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s14.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s16.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name
="genius-gel-function-list-polynomials"></a>Polynomials</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-AddPoly"></a>AddPoly</span></dt><dd><pre
class="synopsis">AddPoly (p1,p2)</pre><p>Add two polynomials (vectors).</p></dd><dt><span class="term"><a
name="gel-function-DividePoly"></a>DividePoly</span></dt><dd><pre class="synopsis">DividePoly
(p,q,&r)</pre><p>Divide two polynomials (as vectors) using long division.
+ Returns the quotient
+ of the two polynomials. The optional argument <code class="varname">r</code>
+ is used to return the remainder. The remainder will have lower
+ degree than <code class="varname">q</code>.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/PolynomialLongDivision"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-IsPoly"></a>IsPoly</span></dt><dd><pre
class="synopsis">IsPoly (p)</pre><p>Check if a vector is usable as a polynomial.</p></dd><dt><span
class="term"><a name="gel-function-MultiplyPoly"></a>MultiplyPoly</span></dt><dd><pre
class="synopsis">MultiplyPoly (p1,p2)</pre><p>Multiply two polynomials (as vectors).</p></dd><dt><span
class="term"><a name="gel-function-NewtonsMethodPoly"></a>NewtonsMethodPoly</span></dt><dd><pre
class="synopsis">NewtonsMethodPoly (poly,guess,epsilon,maxn)</pre><p>Find a root of a polynomial using
Newton's method. <code class="varname">poly</code> is
+ the polynomial as a vector and <code class="varname">guess</code> is the initial
+ guess. The function returns after two successive values are
+ within <code class="varname">epsilon</code> of each other, or after <code
class="varname">maxn</code> tries, in which case the function returns <code class="constant">null</code>
indicating failure.
+ </p><p>
+ See also <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a>.
+ </p><p>
+ Example to find the square root of 10:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethodPoly([-10,0,1],3,10^-10,100)</code></strong>
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Take second polynomial (as vector)
derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Take polynomial (as vector) derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Make function out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Make string out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Subtract two polynomials (as vectors).
</p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Trim zeros from a polynomial (as vector).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Set
Theory</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL
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class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"
<a name="genius-gel-function-list-set-theory"></a>Set Theory</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Intersection"></a>Intersection</span></dt><dd><pre class="synopsis">Intersection
(X,Y)</pre><p>Returns a set theoretic intersection of X and Y (X and Y are vectors pretending to be
sets).</p></dd><dt><span class="term"><a name="gel-function-IsIn"></a>IsIn</span></dt><dd><pre
class="synopsis">IsIn (x,X)</pre><p>Returns <code class="constant">true</code> if the element x is in the
set X (where X is a vector pretending to be a set).</p></dd><dt><span class="term"><a
name="gel-function-IsSubset"></a>IsSubset</span></dt><dd><pre class="synopsis">IsSubset (X,
Y)</pre><p>Returns <code class="constant">true</code> if X is a subset of Y (X and Y are vectors pretending
to be sets).</p></dd><dt><span class="term"><a name="gel-function-MakeSet"></a>MakeSet</span></dt><dd><pre
class="synopsis">MakeSet (X)</pre>
<p>Returns a vector where every element of X appears only once.</p></dd><dt><span class="term"><a
name="gel-function-SetMinus"></a>SetMinus</span></dt><dd><pre class="synopsis">SetMinus (X,Y)</pre><p>Returns
a set theoretic difference X-Y (X and Y are vectors pretending to be sets).</p></dd><dt><span class="term"><a
name="gel-function-Union"></a>Union</span></dt><dd><pre class="synopsis">Union (X,Y)</pre><p>Returns a set
theoretic union of X and Y (X and Y are vectors pretending to be sets).</p></dd></dl></div></div><div
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width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><
td width="40%" align="right" valign="top"> Commutative Algebra</td></tr></table></div></body></html>
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Commutative
Algebra</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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align="center">Commutative Algebra</th></tr><tr><td width="20%" align="left"><a accesskey="p"
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width="20%" align="right"> <a accesskey="n" href="ch11s18.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="c
lear: both"><a name="genius-gel-function-list-commutative-algebra"></a>Commutative
Algebra</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-MacaulayBound"></a>MacaulayBound</span></dt><dd><pre class="synopsis">MacaulayBound
(c,d)</pre><p>For a Hilbert function that is c for degree d, given the Macaulay bound for the Hilbert
function of degree d+1 (The c^<d> operator from Green's proof).</p><p>Version 1.0.15
onwards.</p></dd><dt><span class="term"><a
name="gel-function-MacaulayLowerOperator"></a>MacaulayLowerOperator</span></dt><dd><pre
class="synopsis">MacaulayLowerOperator (c,d)</pre><p>The c_<d> operator from Green's proof of
Macaulay's Theorem.</p><p>Version 1.0.15 onwards.</p></dd><dt><span class="term"><a
name="gel-function-MacaulayRep"></a>MacaulayRep</span></dt><dd><pre class="synopsis">MacaulayRep
(c,d)</pre><p>Return the dth Macaulay representation of a positive integer c.</p><p>Version
1.0.15 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s16.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscellaneous</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s17.html" title="Commutative
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summary="Navigation header"><tr><th colspan="3" align="center">Miscellaneous</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Prev</a> </td><th width="60%" align="center">Chapter 11.
List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
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class="titlepage"><div><div><h2 class="title" style
="clear: both"><a name="genius-gel-function-list-miscellaneous"></a>Miscellaneous</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a vector of ASCII values.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string t
o a vector of 0-based alphabet values (positions in the alphabet string), -1's for unknown
letters.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
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</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Symbolic
Operations</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL
functions"><link rel="prev" href="ch11s18.html" title="Miscellaneous"><link rel="next" href="ch11s20.html"
title="Plotting"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Symbolic Operations</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s18.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s20.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="cle
ar: both"><a name="genius-gel-function-list-symbolic"></a>Symbolic Operations</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-SymbolicDerivative"></a>SymbolicDerivative</span></dt><dd><pre
class="synopsis">SymbolicDerivative (f)</pre><p>Attempt to symbolically differentiate the function f, where f
is a function of one variable.</p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SymbolicDerivative(sin)</code></strong>
+= (`(x)=cos(x))
+<code class="prompt">genius></code> <strong
class="userinput"><code>SymbolicDerivative(`(x)=7*x^2)</code></strong>
+= (`(x)=(7*(2*x)))
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SymbolicDerivativeTry"></a>SymbolicDerivativeTry</span></dt><dd><pre
class="synopsis">SymbolicDerivativeTry (f)</pre><p>Attempt to symbolically differentiate the function f,
where f is a function of one variable, returns <code class="constant">null</code> if unsuccessful but is
silent.
+ (See <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>)
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SymbolicNthDerivative"></a>SymbolicNthDerivative</span></dt><dd><pre
class="synopsis">SymbolicNthDerivative (f,n)</pre><p>Attempt to symbolically differentiate a function n times.
+ (See <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>)
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SymbolicNthDerivativeTry"></a>SymbolicNthDerivativeTry</span></dt><dd><pre
class="synopsis">SymbolicNthDerivativeTry (f,n)</pre><p>Attempt to symbolically differentiate a function n
times quietly and return <code class="constant">null</code> on failure
+ (See <a class="link" href="ch11s19.html#gel-function-SymbolicNthDerivative"><code
class="function">SymbolicNthDerivative</code></a>)
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SymbolicTaylorApproximationFunction"></a>SymbolicTaylorApproximationFunction</span></dt><dd><pre
class="synopsis">SymbolicTaylorApproximationFunction (f,x0,n)</pre><p>Attempt to construct the Taylor
approximation function around x0 to the nth degree.
+ (See <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>)
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s18.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s20.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Miscellaneous </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Plotting</td></tr></table></div></body></html>
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@@ -0,0 +1,445 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Plotting</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
href="ch11s19.html" title="Symbolic Operations"><link rel="next" href="ch12.html" title="Chapter 12. Example
Programs in GEL"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Plotting</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s19.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td
width="20%" align="right"> <a accesskey="n" href="ch12.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" sty
le="clear: both"><a name="genius-gel-function-list-plotting"></a>Plotting</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ExportPlot"></a>ExportPlot</span></dt><dd><pre class="synopsis">ExportPlot
(file,type)</pre><pre class="synopsis">ExportPlot (file)</pre><p>
+ Export the contents of the plotting window to a file.
+ The type is a string that specifies the file type to
+ use, "png", "eps", or "ps". If the type is not
+ specified, then it is taken to be the extension, in
+ which case the extension must be ".png", ".eps", or ".ps".
+ </p><p>
+ Note that files are overwritten without asking.
+ </p><p>
+ On successful export, true is returned. Otherwise
+ error is printed and exception is raised.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ExportPlot("file.png")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>ExportPlot("/directory/file","eps")</code></strong>
+</pre><p>
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LinePlot"></a>LinePlot</span></dt><dd><pre class="synopsis">LinePlot
(func1,func2,func3,...)</pre><pre class="synopsis">LinePlot (func1,func2,func3,x1,x2)</pre><pre
class="synopsis">LinePlot (func1,func2,func3,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlot
(func1,func2,func3,[x1,x2])</pre><pre class="synopsis">LinePlot (func1,func2,func3,[x1,x2,y1,y2])</pre><p>
+ Plot a function (or several functions) with a line.
+ First (up to 10) arguments are functions, then optionally
+ you can specify the limits of the plotting window as
+ <code class="varname">x1</code>, <code class="varname">x2</code>,
+ <code class="varname">y1</code>, <code class="varname">y2</code>. If limits are not
+ specified, then the currently set limits apply
+ (See <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>)
+ If the y limits are not specified, then the functions are computed and then the maxima and minima
+ are used.
+ </p><p>
+ The parameter
+ <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a>
+ controls the drawing of the legend.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlot(sin,cos)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlot(`(x)=x^2,-1,1,0,1)</code></strong>
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre class="synopsis">LinePlotClear
()</pre><p>
+ Show the line plot window and clear out functions and any other
+ lines that were drawn.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (func,...)</pre><pre class="synopsis">LinePlotCParametric
(func,t1,t2,tinc)</pre><pre class="synopsis">LinePlotCParametric (func,t1,t2,tinc,x1,x2,y1,y2)</pre><p>
+ Plot a parametric complex valued function with a line. First comes
+the function that returns <code class="computeroutput">x+iy</code>,
+then optionally the <code class="varname">t</code> limits as <strong
class="userinput"><code>t1,t2,tinc</code></strong>, then
+optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.
+ </p><p>
+ If limits are not
+ specified, then the currently set limits apply
+ (See <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).
+ If instead the string "fit" is given for the x and y limits, then the limits are the maximum
extent of
+ the graph
+ </p><p>
+ The parameter
+ <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a>
+ controls the drawing of the legend.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>
+ Draw a line from <code class="varname">x1</code>,<code class="varname">y1</code> to
+ <code class="varname">x2</code>,<code class="varname">y2</code>.
+ <code class="varname">x1</code>,<code class="varname">y1</code>,
+ <code class="varname">x2</code>,<code class="varname">y2</code> can be replaced by an
+ <code class="varname">n</code> by 2 matrix for a longer polyline.
+ Alternatively the vector <code class="varname">v</code> may be a column vector of complex numbers,
+ that is an <code class="varname">n</code> by 1 matrix and each complex number is then
+ considered a point in the plane.
+ </p><p>
+ Extra parameters can be added to specify line color, thickness,
+ arrows, the plotting window, or legend.
+ You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
+ <strong class="userinput"><code>"thickness"</code></strong>,
+ <strong class="userinput"><code>"window"</code></strong>,
+ <strong class="userinput"><code>"arrow"</code></strong>, or <strong
class="userinput"><code>"legend"</code></strong>, and after it specify
+ the color, the thickness, the window
+ as 4-vector, type of arrow, or the legend. (Arrow and window are from version 1.0.6 onwards.)
+ </p><p>
+ If the line is to be treated as a filled polygon, filled with the given color, you
+ can specify the argument <strong class="userinput"><code>"filled"</code></strong>. Since version
1.0.22 onwards.
+ </p><p>
+ The color should be either a string indicating the common English word for the color
+ that GTK will recognize such as
+ <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
etc...
+ Alternatively the color can be specified in RGB format as
+ <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong>, or
+ <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, where the r, g, or b are hex
digits of the red, green, and blue
+ components of the color. Finally, since version 1.0.18, the color
+ can also be specified as a real vector specifying the red green and
+ blue components where the components are between 0 and 1, e.g. <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.
+ </p><p>
+ The window should be given as usual as <strong
class="userinput"><code>[x1,x2,y1,y2]</code></strong>, or
+ alternatively can be given as a string
+ <strong class="userinput"><code>"fit"</code></strong> in which case,
+ the x range will be set precisely and the y range will be set with
+ five percent borders around the line.
+ </p><p>
+ Arrow specification should be
+ <strong class="userinput"><code>"origin"</code></strong>,
+ <strong class="userinput"><code>"end"</code></strong>,
+ <strong class="userinput"><code>"both"</code></strong>, or
+ <strong class="userinput"><code>"none"</code></strong>.
+ </p><p>
+ Finally, legend should be a string that can be used as the legend in the
+ graph. That is, if legends are being printed.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
+</pre><p>
+ </p><p>
+ Unlike many other functions that do not care if they take a
+ column or a row vector, if specifying points as a vector of
+ complex values, due to possible ambiguities, it must always
+ be given as a column vector.
+ </p><p>
+ Specifying <code class="varname">v</code> as a column vector of complex numbers is
+ implemented from version 1.0.22 and onwards.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints (v,...)</pre><p>
+ Draw a point at <code class="varname">x</code>,<code class="varname">y</code>.
+ The input can be an <code class="varname">n</code> by 2 matrix
+ for <code class="varname">n</code> different points. This function has essentially the same
+ input as <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>.
+ Alternatively the vector <code class="varname">v</code> may be a column vector of complex
numbers,
+ that is an <code class="varname">n</code> by 1 matrix and each complex number is then
+ considered a point in the plane.
+ </p><p>
+ Extra parameters can be added to specify color, thickness,
+ the plotting window, or legend.
+ You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
+ <strong class="userinput"><code>"thickness"</code></strong>,
+ <strong class="userinput"><code>"window"</code></strong>,
+ or <strong class="userinput"><code>"legend"</code></strong>, and after it specify
+ the color, the thickness, the window
+ as 4-vector, or the legend.
+ </p><p>
+ The color should be either a string indicating the common English word for the color
+ that GTK will recognize such as
+ <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
etc...
+ Alternatively the color can be specified in RGB format as
+ <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong>, or
+ <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, where the r, g, or b are hex
digits of the red, green, and blue
+ components of the color. Finally the color can also be specified as a real vector specifying the
red green
+ and blue components where the components are between 0 and 1.
+ </p><p>
+ The window should be given as usual as <strong
class="userinput"><code>[x1,x2,y1,y2]</code></strong>, or
+ alternatively can be given as a string
+ <strong class="userinput"><code>"fit"</code></strong> in which case,
+ the x range will be set precisely and the y range will be set with
+ five percent borders around the line.
+ </p><p>
+ Finally, legend should be a string that can be used as the legend in the
+ graph. That is, if legends are being printed.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
+</pre><p>
+ </p><p>
+ Unlike many other functions that do not care if they take a
+ column or a row vector, if specifying points as a vector of
+ complex values, due to possible ambiguities, it must always
+ be given as a column vector. Therefore, notice in the
+ last example the transpose of the vector <strong class="userinput"><code>0:6</code></strong>
+ to make it into a column vector.
+ </p><p>
+ Available from version 1.0.18 onwards. Specifying
+ <code class="varname">v</code> as a column vector of complex numbers is
+ implemented from version 1.0.22 and onwards.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotMouseLocation"></a>LinePlotMouseLocation</span></dt><dd><pre
class="synopsis">LinePlotMouseLocation ()</pre><p>
+ Returns a row vector of a point on the line plot corresponding to
+ the current mouse location. If the line plot is not visible,
+ then prints an error and returns <code class="constant">null</code>.
+ In this case you should run
+ <a class="link" href="ch11s20.html#gel-function-LinePlot"><code
class="function">LinePlot</code></a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotClear"><code
class="function">LinePlotClear</code></a>
+ to put the graphing window into the line plot mode.
+ See also
+ <a class="link" href="ch11s20.html#gel-function-LinePlotWaitForClick"><code
class="function">LinePlotWaitForClick</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotParametric"></a>LinePlotParametric</span></dt><dd><pre
class="synopsis">LinePlotParametric (xfunc,yfunc,...)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,[x1,x2,y1,y2])</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,"fit")</pre><p>
+ Plot a parametric function with a line. First come the functions
+for <code class="varname">x</code> and <code class="varname">y</code> then optionally the <code
class="varname">t</code> limits as <strong class="userinput"><code>t1,t2,tinc</code></strong>, then
optionally the
+limits as <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.
+ </p><p>
+ If x and y limits are not
+ specified, then the currently set limits apply
+ (See <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).
+ If instead the string "fit" is given for the x and y limits, then the limits are the maximum
extent of
+ the graph
+ </p><p>
+ The parameter
+ <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a>
+ controls the drawing of the legend.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinePlotWaitForClick"></a>LinePlotWaitForClick</span></dt><dd><pre
class="synopsis">LinePlotWaitForClick ()</pre><p>
+ If in line plot mode, waits for a click on the line plot window
+ and returns the location of the click as a row vector.
+ If the window is closed
+ the function returns immediately with <code class="constant">null</code>.
+ If the window is not in line plot mode, it is put in it and shown
+ if not shown.
+ See also
+ <a class="link" href="ch11s20.html#gel-function-LinePlotMouseLocation"><code
class="function">LinePlotMouseLocation</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasFreeze"></a>PlotCanvasFreeze</span></dt><dd><pre
class="synopsis">PlotCanvasFreeze ()</pre><p>
+ Freeze drawing of the canvas plot temporarily. Useful if you need to draw a bunch of
elements
+ and want to delay drawing everything to avoid flicker in an animation. After everything
+ has been drawn you should call <a class="link"
href="ch11s20.html#gel-function-PlotCanvasThaw"><code class="function">PlotCanvasThaw</code></a>.
+ </p><p>
+ The canvas is always thawed after end of any execution, so it will never remain frozen.
The moment
+ a new command line is shown for example the plot canvas is thawed automatically. Also note
that
+ calls to freeze and thaw may be safely nested.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasThaw"></a>PlotCanvasThaw</span></dt><dd><pre class="synopsis">PlotCanvasThaw
()</pre><p>
+ Thaw the plot canvas frozen by
+ <a class="link" href="ch11s20.html#gel-function-PlotCanvasFreeze"><code
class="function">PlotCanvasFreeze</code></a>
+ and redraw the canvas immediately. The canvas is also always thawed after end of execution
+ of any program.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PlotWindowPresent"></a>PlotWindowPresent</span></dt><dd><pre
class="synopsis">PlotWindowPresent ()</pre><p>
+ Show and raise the plot window, creating it if necessary.
+ Normally the window is created when one of the plotting
+ functions is called, but it is not always raised if it
+ happens to be below other windows. So this function is
+ good to call in scripts where the plot window might have
+ been created before, and by now is hidden behind the
+ console or other windows.
+ </p><p>Version 1.0.19 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldClearSolutions"></a>SlopefieldClearSolutions</span></dt><dd><pre
class="synopsis">SlopefieldClearSolutions ()</pre><p>
+ Clears the solutions drawn by the
+ <a class="link" href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>
+ function.
+ </p></dd><dt><span class="term"><a
name="gel-function-SlopefieldDrawSolution"></a>SlopefieldDrawSolution</span></dt><dd><pre
class="synopsis">SlopefieldDrawSolution (x, y, dx)</pre><p>
+ When a slope field plot is active, draw a solution with
+ the specified initial condition. The standard
+ Runge-Kutta method is used with increment <code class="varname">dx</code>.
+ Solutions stay on the graph until a different plot is shown or until
+ you call
+ <a class="link" href="ch11s20.html#gel-function-SlopefieldClearSolutions"><code
class="function">SlopefieldClearSolutions</code></a>.
+ You can also use the graphical interface to draw solutions and specify
+ initial conditions with the mouse.
+ </p></dd><dt><span class="term"><a
name="gel-function-SlopefieldPlot"></a>SlopefieldPlot</span></dt><dd><pre class="synopsis">SlopefieldPlot
(func)</pre><pre class="synopsis">SlopefieldPlot (func,x1,x2,y1,y2)</pre><p>
+ Plot a slope field. The function <code class="varname">func</code>
+ should take two real numbers <code class="varname">x</code>
+ and <code class="varname">y</code>, or a single complex
+ number.
+ Optionally you can specify the limits of the plotting window as
+ <code class="varname">x1</code>, <code class="varname">x2</code>,
+ <code class="varname">y1</code>, <code class="varname">y2</code>. If limits are not
+ specified, then the currently set limits apply
+ (See <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).
+ </p><p>
+ The parameter
+ <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a>
+ controls the drawing of the legend.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</code></strong>
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-SurfacePlot"></a>SurfacePlot</span></dt><dd><pre class="synopsis">SurfacePlot
(func)</pre><pre class="synopsis">SurfacePlot (func,x1,x2,y1,y2,z1,z2)</pre><pre class="synopsis">SurfacePlot
(func,x1,x2,y1,y2)</pre><pre class="synopsis">SurfacePlot (func,[x1,x2,y1,y2,z1,z2])</pre><pre
class="synopsis">SurfacePlot (func,[x1,x2,y1,y2])</pre><p>
+ Plot a surface function that takes either two arguments or a complex number. First comes the
function then optionally limits as <code class="varname">x1</code>, <code class="varname">x2</code>,
+ <code class="varname">y1</code>, <code class="varname">y2</code>,
+ <code class="varname">z1</code>, <code class="varname">z2</code>. If limits are not
+ specified, then the currently set limits apply
+ (See <a class="link" href="ch11s03.html#gel-function-SurfacePlotWindow"><code
class="function">SurfacePlotWindow</code></a>).
+ Genius can only plot a single surface function at this time.
+ </p><p>
+ If the z limits are not specified then the maxima and minima of the function are used.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(|sin|,-1,1,-1,1,0,1.5)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(`(x,y)=x^2+y,-1,1,-1,1,-2,2)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(`(z)=|z|^2,-1,1,-1,1,0,2)</code></strong>
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotClear"></a>SurfacePlotClear</span></dt><dd><pre
class="synopsis">SurfacePlotClear ()</pre><p>
+ Show the surface plot window and clear out functions and any other
+ lines that were drawn.
+ </p><p>
+ Available in version 1.0.19 and onwards.
+ </p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotData"></a>SurfacePlotData</span></dt><dd><pre class="synopsis">SurfacePlotData
(data)</pre><pre class="synopsis">SurfacePlotData (data,label)</pre><pre class="synopsis">SurfacePlotData
(data,x1,x2,y1,y2,z1,z2)</pre><pre class="synopsis">SurfacePlotData (data,label,x1,x2,y1,y2,z1,z2)</pre><pre
class="synopsis">SurfacePlotData (data,[x1,x2,y1,y2,z1,z2])</pre><pre class="synopsis">SurfacePlotData
(data,label,[x1,x2,y1,y2,z1,z2])</pre><p>
+ Plot a surface from data. The data is an n by 3 matrix whose
+ rows are the x, y and z coordinates. The data can also be
+ simply a vector whose length is a multiple of 3 and so
+ contains the triples of x, y, z. The data should contain at
+ least 3 points.
+ </p><p>
+ Optionally we can give the label and also optionally the
+ limits. If limits are not given, they are computed from
+ the data, <a class="link" href="ch11s03.html#gel-function-SurfacePlotWindow"><code
class="function">SurfacePlotWindow</code></a>
+ is not used, if you want to use it, pass it in explicitly.
+ If label is not given then empty label is used.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotData([0,0,0;1,0,1;0,1,1;1,1,3])</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>SurfacePlotData(data,"My
data")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotData(data,-1,1,-1,1,0,10)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotData(data,SurfacePlotWindow)</code></strong>
+</pre><p>
+ </p><p>
+ Here's an example of how to plot in polar coordinates,
+ in particular how to plot the function
+ <strong class="userinput"><code>-r^2 * theta</code></strong>:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>d:=null; for r=0 to 1 by 0.1 do for theta=0 to 2*pi by pi/5 do
d=[d;[r*cos(theta),r*sin(theta),-r^2*theta]];</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>SurfacePlotData(d)</code></strong>
+</pre><p>
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDataGrid"></a>SurfacePlotDataGrid</span></dt><dd><pre
class="synopsis">SurfacePlotDataGrid (data,[x1,x2,y1,y2])</pre><pre class="synopsis">SurfacePlotDataGrid
(data,[x1,x2,y1,y2,z1,z2])</pre><pre class="synopsis">SurfacePlotDataGrid
(data,[x1,x2,y1,y2],label)</pre><pre class="synopsis">SurfacePlotDataGrid
(data,[x1,x2,y1,y2,z1,z2],label)</pre><p>
+ Plot a surface from regular rectangular data.
+ The data is given in a n by m matrix where the rows are the
+ x coordinate and the columns are the y coordinate.
+ The x coordinate is divided into equal n-1 subintervals
+ and y coordinate is divided into equal m-1 subintervals.
+ The limits <code class="varname">x1</code> and <code class="varname">x2</code>
+ give the interval on the x-axis that we use, and
+ the limits <code class="varname">y1</code> and <code class="varname">y2</code>
+ give the interval on the y-axis that we use.
+ If the limits <code class="varname">z1</code> and <code class="varname">z2</code>
+ are not given they are computed from the data (to be
+ the extreme values from the data).
+ </p><p>
+ Optionally we can give the label, if label is not given then
+ empty label is used.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid([1,2;3,4],[0,1,0,1])</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(data,[-1,1,-1,1],"My data")</code></strong>
+<code class="prompt">genius></code> <strong class="userinput"><code>d:=null; for i=1 to 20 do for j=1 to
10 do d@(i,j) = (0.1*i-1)^2-(0.1*j)^2;</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(d,[-1,1,0,1],"half a saddle")</code></strong>
+</pre><p>
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>
+ Draw a line from <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> to
+ <code class="varname">x2</code>,<code class="varname">y2</code>,<code class="varname">z2</code>.
+ <code class="varname">x1</code>,<code class="varname">y1</code>,<code class="varname">z1</code>,
+ <code class="varname">x2</code>,<code class="varname">y2</code>,<code class="varname">z2</code>
can be replaced by an
+ <code class="varname">n</code> by 3 matrix for a longer polyline.
+ </p><p>
+ Extra parameters can be added to specify line color, thickness,
+ arrows, the plotting window, or legend.
+ You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
+ <strong class="userinput"><code>"thickness"</code></strong>,
+ <strong class="userinput"><code>"window"</code></strong>,
+ or <strong class="userinput"><code>"legend"</code></strong>, and after it specify
+ the color, the thickness, the window
+ as 6-vector, or the legend.
+ </p><p>
+ The color should be either a string indicating the common English word for the color
+ that GTK will recognize such as
+ <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
etc...
+ Alternatively the color can be specified in RGB format as
+ <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong>, or
+ <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, where the r, g, or b are hex
digits of the red, green, and blue
+ components of the color. Finally, since version 1.0.18, the color
+ can also be specified as a real vector specifying the red green and
+ blue components where the components are between 0 and 1, e.g. <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.
+ </p><p>
+ The window should be given as usual as <strong
class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, or
+ alternatively can be given as a string
+ <strong class="userinput"><code>"fit"</code></strong> in which case,
+ the x range will be set precisely and the y range will be set with
+ five percent borders around the line.
+ </p><p>
+ Finally, legend should be a string that can be used as the legend in the
+ graph. That is, if legends are being printed.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine([0,0,0;1,-1,2;-1,-1,-3])</code></strong>
+</pre><p>
+ </p><p>
+ Available from version 1.0.19 onwards.
+ </p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawPoints"></a>SurfacePlotDrawPoints</span></dt><dd><pre
class="synopsis">SurfacePlotDrawPoints (x,y,z,...)</pre><pre class="synopsis">SurfacePlotDrawPoints
(v,...)</pre><p>
+ Draw a point at <code class="varname">x</code>,<code class="varname">y</code>,<code
class="varname">z</code>.
+ The input can be an <code class="varname">n</code> by 3 matrix
+ for <code class="varname">n</code> different points. This function has essentially the same
+ input as <a class="link"
href="ch11s20.html#gel-function-SurfacePlotDrawLine">SurfacePlotDrawLine</a>.
+ </p><p>
+ Extra parameters can be added to specify line color, thickness,
+ the plotting window, or legend.
+ You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
+ <strong class="userinput"><code>"thickness"</code></strong>,
+ <strong class="userinput"><code>"window"</code></strong>,
+ or <strong class="userinput"><code>"legend"</code></strong>, and after it specify
+ the color, the thickness, the window
+ as 6-vector, or the legend.
+ </p><p>
+ The color should be either a string indicating the common English word for the color
+ that GTK will recognize such as
+ <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
etc...
+ Alternatively the color can be specified in RGB format as
+ <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong>, or
+ <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, where the r, g, or b are hex
digits of the red, green, and blue
+ components of the color. Finally the color can also be specified as a real vector specifying the
red green
+ and blue components where the components are between 0 and 1.
+ </p><p>
+ The window should be given as usual as <strong
class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, or
+ alternatively can be given as a string
+ <strong class="userinput"><code>"fit"</code></strong> in which case,
+ the x range will be set precisely and the y range will be set with
+ five percent borders around the line.
+ </p><p>
+ Finally, legend should be a string that can be used as the legend in the
+ graph. That is, if legends are being printed.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints([0,0,0;1,-1,2;-1,-1,1])</code></strong>
+</pre><p>
+ </p><p>
+ Available from version 1.0.19 onwards.
+ </p></dd><dt><span class="term"><a
name="gel-function-VectorfieldClearSolutions"></a>VectorfieldClearSolutions</span></dt><dd><pre
class="synopsis">VectorfieldClearSolutions ()</pre><p>
+ Clears the solutions drawn by the
+ <a class="link" href="ch11s20.html#gel-function-VectorfieldDrawSolution"><code
class="function">VectorfieldDrawSolution</code></a>
+ function.
+ </p><p>Version 1.0.6 onwards.</p></dd><dt><span class="term"><a
name="gel-function-VectorfieldDrawSolution"></a>VectorfieldDrawSolution</span></dt><dd><pre
class="synopsis">VectorfieldDrawSolution (x, y, dt, tlen)</pre><p>
+ When a vector field plot is active, draw a solution with
+ the specified initial condition. The standard
+ Runge-Kutta method is used with increment <code class="varname">dt</code>
+ for an interval of length <code class="varname">tlen</code>.
+ Solutions stay on the graph until a different plot is shown or until
+ you call
+ <a class="link" href="ch11s20.html#gel-function-VectorfieldClearSolutions"><code
class="function">VectorfieldClearSolutions</code></a>.
+ You can also use the graphical interface to draw solutions and specify
+ initial conditions with the mouse.
+ </p><p>Version 1.0.6 onwards.</p></dd><dt><span class="term"><a
name="gel-function-VectorfieldPlot"></a>VectorfieldPlot</span></dt><dd><pre class="synopsis">VectorfieldPlot
(funcx, funcy)</pre><pre class="synopsis">VectorfieldPlot (funcx, funcy, x1, x2, y1, y2)</pre><p>
+ Plot a two dimensional vector field. The function
+ <code class="varname">funcx</code>
+ should be the dx/dt of the vectorfield and the function
+ <code class="varname">funcy</code> should be the dy/dt of the vectorfield.
+ The functions
+ should take two real numbers <code class="varname">x</code>
+ and <code class="varname">y</code>, or a single complex
+ number. When the parameter
+ <a class="link" href="ch11s03.html#gel-function-VectorfieldNormalized"><code
class="function">VectorfieldNormalized</code></a>
+ is <code class="constant">true</code>, then the magnitude of the vectors is normalized. That is,
only
+ the direction and not the magnitude is shown.
+ </p><p>
+ Optionally you can specify the limits of the plotting window as
+ <code class="varname">x1</code>, <code class="varname">x2</code>,
+ <code class="varname">y1</code>, <code class="varname">y2</code>. If limits are not
+ specified, then the currently set limits apply
+ (See <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).
+ </p><p>
+ The parameter
+ <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a>
+ controls the drawing of the legend.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>VectorfieldPlot(`(x,y)=x^2-y, `(x,y)=y^2-x, -1, 1, -1, 1)</code></strong>
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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mple-programs"></a>Chapter 12. Example Programs in GEL</h1></div></div></div><p>
+Here is a function that calculates factorials:
+</p><pre class="programlisting">function f(x) = if x <= 1 then 1 else (f(x-1)*x)
+</pre><p>
+ </p><p>
+With indentation it becomes:
+</p><pre class="programlisting">function f(x) = (
+ if x <= 1 then
+ 1
+ else
+ (f(x-1)*x)
+)
+</pre><p>
+ </p><p>
+This is a direct port of the factorial function from the <span class="application">bc</span> manpage. The
syntax seems similar to <span class="application">bc</span>, but different in that in GEL, the last
expression is the one that is returned. Using the <code class="literal">return</code> function instead, it
would be:
+</p><pre class="programlisting">function f(x) = (
+ if (x <= 1) then return (1);
+ return (f(x-1) * x)
+)
+</pre><p>
+ </p><p>
+By far the easiest way to define a factorial function would be using
+the product loop as follows. This is not only the shortest and fastest,
+but also probably the most readable version.
+</p><pre class="programlisting">function f(x) = prod k=1 to x do k
+</pre><p>
+ </p><p>
+Here is a larger example, this basically redefines the internal
+<a class="link" href="ch11s09.html#gel-function-ref"><code class="function">ref</code></a> function to
calculate the row echelon form of a
+matrix. The function <code class="function">ref</code> is built in and much faster,
+but this example demonstrates some of the more complex features of GEL.
+</p><pre class="programlisting"># Calculate the row-echelon form of a matrix
+function MyOwnREF(m) = (
+ if not IsMatrix(m) or not IsValueOnly(m) then
+ (error("MyOwnREF: argument not a value only matrix");bailout);
+ s := min(rows(m), columns(m));
+ i := 1;
+ d := 1;
+ while d <= s and i <= columns(m) do (
+
+ # This just makes the anchor element non-zero if at
+ # all possible
+ if m@(d,i) == 0 then (
+ j := d+1;
+ while j <= rows(m) do (
+ if m@(j,i) == 0 then
+ (j=j+1;continue);
+ a := m@(j,);
+ m@(j,) := m@(d,);
+ m@(d,) := a;
+ j := j+1;
+ break
+ )
+ );
+ if m@(d,i) == 0 then
+ (i:=i+1;continue);
+
+ # Here comes the actual zeroing of all but the anchor
+ # element rows
+ j := d+1;
+ while j <= rows(m)) do (
+ if m@(j,i) != 0 then (
+ m@(j,) := m@(j,)-(m@(j,i)/m@(d,i))*m@(d,)
+ );
+ j := j+1
+ );
+ m@(d,) := m@(d,) * (1/m@(d,i));
+ d := d+1;
+ i := i+1
+ );
+ m
+)
+</pre><p>
+ </p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%"
align="left"><a accesskey="p" href="ch11s20.html">Prev</a> </td><td width="20%" align="center"> </td><td
width="40%" align="right"> <a accesskey="n" href="ch13.html">Next</a></td></tr><tr><td width="40%"
align="left" valign="top">Plotting </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Chapter 13.
Settings</td></tr></table></div></body></html>
diff --git a/help/C/ch13.html b/help/C/ch13.html
new file mode 100644
index 0000000..82415a5
--- /dev/null
+++ b/help/C/ch13.html
@@ -0,0 +1,73 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 13.
Settings</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="index.html" title="Genius Manual"><link
rel="prev" href="ch12.html" title="Chapter 12. Example Programs in GEL"><link rel="next" href="ch13s02.html"
title="Precision"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Chapter 13. Settings</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch12.html">Prev</a> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a
accesskey="n" href="ch13s02.html">Next</a></td></tr></table><hr></div><div class="chapter"><div
class="titlepage"><div><div><h1 class="title"><a name="genius-prefs"></a>Chapter 1
3. Settings</h1></div></div></div><div class="toc"><p><b>Table of Contents</b></p><dl class="toc"><dt><span
class="sect1"><a href="ch13.html#genius-prefs-output">Output</a></span></dt><dt><span class="sect1"><a
href="ch13s02.html">Precision</a></span></dt><dt><span class="sect1"><a
href="ch13s03.html">Terminal</a></span></dt><dt><span class="sect1"><a
href="ch13s04.html">Memory</a></span></dt></dl></div><p>
+ To configure <span class="application">Genius Mathematics Tool</span>, choose
+ <span class="guimenu">Settings</span> → <span class="guimenuitem">Preferences</span>.
+ There are several basic parameters provided by the calculator in addition
+ to the ones provided by the standard library. These control how the
+ calculator behaves.
+ </p><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Changing
Settings with GEL</h3><p>
+ Many of the settings in Genius are simply global variables, and can
+ be evaluated and assigned to in the same way as normal variables. See
+ <a class="xref" href="ch05s02.html" title="Using Variables">the section called “Using Variables”</a>
about evaluating and assigning
+ to variables, and <a class="xref" href="ch11s03.html" title="Parameters">the section called
“Parameters”</a> for
+ a list of settings that can be modified in this way.
+ </p><p>
+As an example, you can set the maximum number of digits in a result to 12 by typing:
+</p><pre class="programlisting">MaxDigits = 12
+</pre><p>
+ </p></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-prefs-output"></a>Output</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term">
+ <span class="guilabel">Maximum digits to output</span>
+ </span></dt><dd><p>The maximum digits in a result (<a class="link"
href="ch11s03.html#gel-function-MaxDigits"><code class="function">MaxDigits</code></a>)</p></dd><dt><span
class="term">
+ <span class="guilabel">Results as floats</span>
+ </span></dt><dd><p>If the results should be always printed as floats (<a class="link"
href="ch11s03.html#gel-function-ResultsAsFloats"><code
class="function">ResultsAsFloats</code></a>)</p></dd><dt><span class="term">
+ <span class="guilabel">Floats in scientific notation</span>
+ </span></dt><dd><p>If floats should be in scientific notation (<a class="link"
href="ch11s03.html#gel-function-ScientificNotation"><code
class="function">ScientificNotation</code></a>)</p></dd><dt><span class="term">
+ <span class="guilabel">Always print full expressions</span>
+ </span></dt><dd><p>Should we print out full expressions for non-numeric return values (longer than a
line) (<a class="link" href="ch11s03.html#gel-function-FullExpressions"><code
class="function">FullExpressions</code></a>)</p></dd><dt><span class="term">
+ <span class="guilabel">Use mixed fractions</span>
+ </span></dt><dd><p>If fractions should be printed as mixed fractions such as "1 1/3" rather than
"4/3". (<a class="link" href="ch11s03.html#gel-function-MixedFractions"><code
class="function">MixedFractions</code></a>)</p></dd><dt><span class="term">
+ <span class="guilabel">Display 0.0 when floating point number is less than 10^-x (0=never
chop)</span>
+ </span></dt><dd><p>How to chop output. But only when other numbers nearby are large.
+ See the documentation of the parameter
+ <a class="link" href="ch11s03.html#gel-function-OutputChopExponent"><code
class="function">OutputChopExponent</code></a>. </p></dd><dt><span class="term">
+ <span class="guilabel">Only chop numbers when another number is greater than 10^-x</span>
+ </span></dt><dd><p>When to chop output. This is set by the parameter <a class="link"
href="ch11s03.html#gel-function-OutputChopWhenExponent"><code
class="function">OutputChopWhenExponent</code></a>.
+ See the documentation of the parameter
+ <a class="link" href="ch11s03.html#gel-function-OutputChopExponent"><code
class="function">OutputChopExponent</code></a>. </p></dd><dt><span class="term">
+ <span class="guilabel">Remember output settings across sessions</span>
+ </span></dt><dd><p>Should the output settings in the <span class="guilabel">Number/Expression output
options</span> frame
+ be remembered for next session. Does not apply to the <span class="guilabel">Error/Info output
options</span> frame.</p><p>
+ If unchecked,
+ either the default or any previously saved settings are used each time Genius starts
+ up. Note that
+ settings are saved at the end of the session, so if you wish to change the defaults
+ check this box, restart <span class="application">Genius Mathematics Tool</span> and then uncheck
it again.
+ </p></dd><dt><span class="term">
+ <span class="guilabel">Display errors in a dialog</span>
+ </span></dt><dd><p>If set the errors will be displayed in a separate dialog, if
+ unset the errors will be printed on the console.</p></dd><dt><span class="term">
+ <span class="guilabel">Display information messages in a dialog</span>
+ </span></dt><dd><p>If set the information messages will be displayed in a separate
+ dialog, if unset the information messages will be printed on the
+ console.</p></dd><dt><span class="term">
+ <span class="guilabel">Maximum errors to display</span>
+ </span></dt><dd><p>
+ The maximum number of errors to return on one evaluation
+ (<a class="link" href="ch11s03.html#gel-function-MaxErrors"><code
class="function">MaxErrors</code></a>). If you set this to 0 then
+ all errors are always returned. Usually if some loop causes
+ many errors, then it is unlikely that you will be able to make
+ sense out of more than a few of these, so seeing a long list
+ of errors is usually not helpful.
+ </p></dd></dl></div><p>
+ In addition to these preferences, there are some preferences that can
+ only be changed by setting them in the workspace console. For others
+ that may affect the output see <a class="xref" href="ch11s03.html" title="Parameters">the section
called “Parameters”</a>.
+ </p><div class="variablelist"><dl class="variablelist"><dt><span class="term">
+ <code class="function">IntegerOutputBase</code>
+ </span></dt><dd><p>The base that will be used to output integers</p></dd><dt><span class="term">
+ <code class="function">OutputStyle</code>
+ </span></dt><dd><p>A string, can be <code class="literal">"normal"</code>,
+<code class="literal">"latex"</code>, <code class="literal">"mathml"</code> or
+<code class="literal">"troff"</code> and it will affect how matrices (and perhaps other
+stuff) is printed, useful for pasting into documents. Normal style is the
+default human readable printing style of <span class="application">Genius Mathematics Tool</span>. The
other styles are for
+typesetting in LaTeX, MathML (XML), or in Troff.</p></dd></dl></div></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch12.html">Prev</a> </td><td width="20%" align="center"> </td><td width="40%"
align="right"> <a accesskey="n" href="ch13s02.html">Next</a></td></tr><tr><td width="40%" align="left"
valign="top">Chapter 12. Example Programs in GEL </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Home</a></td><td width="40%" align="right" valign="top">
Precision</td></tr></table></div></body></html>
diff --git a/help/C/ch13s02.html b/help/C/ch13s02.html
new file mode 100644
index 0000000..1cdcc87
--- /dev/null
+++ b/help/C/ch13s02.html
@@ -0,0 +1,20 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Precision</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch13.html" title="Chapter 13. Settings"><link rel="prev" href="ch13.html"
title="Chapter 13. Settings"><link rel="next" href="ch13s03.html" title="Terminal"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Precision</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch13.html">Prev</a> </td><th width="60%"
align="center">Chapter 13. Settings</th><td width="20%" align="right"> <a accesskey="n"
href="ch13s03.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-prefs-precision">
</a>Precision</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span
class="term">
+ <span class="guilabel">Floating point precision</span>
+ </span></dt><dd><p>
+ The floating point precision in bits
+ (<a class="link" href="ch11s03.html#gel-function-FloatPrecision"><code
class="function">FloatPrecision</code></a>).
+ Note that changing this only affects newly computed quantities.
+ Old values stored in variables are obviously still in the old
+ precision and if you want to have them more precise you will have
+ to recompute them. Exceptions to this are the system constants
+ such as <a class="link" href="ch11s04.html#gel-function-pi"><code class="function">pi</code></a> or
+ <a class="link" href="ch11s04.html#gel-function-e"><code class="function">e</code></a>.
+ </p></dd><dt><span class="term">
+ <span class="guilabel">Remember precision setting across sessions</span>
+ </span></dt><dd><p>
+ Should the precision setting be remembered for the next session. If unchecked,
+ either the default or any previously saved setting is used each time Genius starts
+ up. Note that
+ settings are saved at the end of the session, so if you wish to change the default
+ check this box, restart genius and then uncheck it again.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch13.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch13.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch13s03.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Chapter 13. Settings
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Terminal</td></tr></table></div></body></html>
diff --git a/help/C/ch13s03.html b/help/C/ch13s03.html
new file mode 100644
index 0000000..6c2a223
--- /dev/null
+++ b/help/C/ch13s03.html
@@ -0,0 +1,11 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Terminal</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch13.html" title="Chapter 13. Settings"><link rel="prev" href="ch13s02.html"
title="Precision"><link rel="next" href="ch13s04.html" title="Memory"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Terminal</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch13s02.html">Prev</a> </td><th width="60%" align="center">Chapter 13.
Settings</th><td width="20%" align="right"> <a accesskey="n"
href="ch13s04.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-prefs-terminal"></a>Terminal
</h2></div></div></div><p>
+ Terminal refers to the console in the work area.
+ </p><div class="variablelist"><dl class="variablelist"><dt><span class="term">
+ <span class="guilabel">Scrollback lines</span>
+ </span></dt><dd><p>Lines of scrollback in the terminal.</p></dd><dt><span class="term">
+ <span class="guilabel">Font</span>
+ </span></dt><dd><p>The font to use on the terminal.</p></dd><dt><span class="term">
+ <span class="guilabel">Black on white</span>
+ </span></dt><dd><p>If to use black on white on the terminal.</p></dd><dt><span class="term">
+ <span class="guilabel">Blinking cursor</span>
+ </span></dt><dd><p>If the cursor in the terminal should blink when the terminal is in focus. This can
sometimes be annoying and it generates idle traffic if you are using Genius
remotely.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch13s02.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch13.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch13s04.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Precision </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Memory</td></tr></table></div></body></html>
diff --git a/help/C/ch13s04.html b/help/C/ch13s04.html
new file mode 100644
index 0000000..b5593c2
--- /dev/null
+++ b/help/C/ch13s04.html
@@ -0,0 +1,20 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Memory</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch13.html" title="Chapter 13. Settings"><link rel="prev" href="ch13s03.html"
title="Terminal"><link rel="next" href="ch14.html" title="Chapter 14. About Genius Mathematics
Tool"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Memory</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch13s03.html">Prev</a>
</td><th width="60%" align="center">Chapter 13. Settings</th><td width="20%" align="right"> <a accesskey="n"
href="ch14.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-pr
efs-memory"></a>Memory</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span
class="term">
+ <span class="guilabel">Maximum number of nodes to allocate</span>
+ </span></dt><dd><p>
+ Internally all data is put onto small nodes in memory. This gives
+ a limit on the maximum number of nodes to allocate for
+ computations. This limit avoids the problem of running out of memory
+ if you do something by mistake that uses too much memory, such
+ as a recursion without end. This could slow your computer and make
+ it hard to even interrupt the program.
+ </p><p>
+ Once the limit is reached, <span class="application">Genius Mathematics Tool</span> asks if you
wish to interrupt
+ the computation or if you wish to continue. If you continue, no
+ limit is applied and it will be possible to run your computer
+ out of memory. The limit will be applied again next time you
+ execute a program or an expression on the Console regardless of how
+ you answered the question.
+ </p><p>
+ Setting the limit to zero means there is no limit to the amount of
+ memory that genius uses.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch13s03.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch13.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch14.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Terminal </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Chapter 14. About <span class="application">Genius Mathematics
Tool</span></td></tr></table></div></body></html>
diff --git a/help/C/ch14.html b/help/C/ch14.html
new file mode 100644
index 0000000..a9bc727
--- /dev/null
+++ b/help/C/ch14.html
@@ -0,0 +1,22 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Chapter 14. About
Genius Mathematics Tool</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Genius Manual"><link rel="up" href="index.html" title="Genius
Manual"><link rel="prev" href="ch13s04.html" title="Memory"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Chapter 14. About <span class="application">Genius Mathematics
Tool</span></th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch13s04.html">Prev</a> </td><th
width="60%" align="center"> </th><td width="20%" align="right"> </td></tr></table><hr></div><div
class="chapter"><div class="titlepage"><div><div><h1 class="title"><a name="genius-about"></a>Chapter 14.
About <span class="application">Genius Mathema
tics Tool</span></h1></div></div></div><p> <span class="application">Genius Mathematics Tool</span> was
written by Jiří (George) Lebl
+(<code class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code>). The
history of <span class="application">Genius Mathematics Tool</span> goes back to late
+1997. It was the first calculator program for GNOME, but it then grew
+beyond being just a desktop calculator. To find more information about
+<span class="application">Genius Mathematics Tool</span>, please visit the <a class="ulink"
href="http://www.jirka.org/genius.html"; target="_top">Genius Web page</a>.
+ </p><p>
+ To report a bug or make a suggestion regarding this application or
+ this manual, send email to me (the author) or post to the mailing
+ list (see the web page).
+ </p><p> This program is distributed under the terms of the GNU
+ General Public license as published by the Free Software
+ Foundation; either version 3 of the License, or (at your option)
+ any later version. A copy of this license can be found at this
+ <a class="ulink" href="http://www.gnu.org/copyleft/gpl.html"; target="_top">link</a>, or in the file
+ COPYING included with the source code of this program. </p><p>Jiří Lebl was during various parts of
the development
+ partially supported for the work by NSF grants DMS 0900885,
+ DMS 1362337,
+ the University of Illinois at Urbana-Champaign,
+ the University of California at San Diego,
+ the University of Wisconsin-Madison, and
+ Oklahoma State University. The software has
+ been used for both teaching and research.</p></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch13s04.html">Prev</a>
</td><td width="20%" align="center"> </td><td width="40%" align="right"> </td></tr><tr><td width="40%"
align="left" valign="top">Memory </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Home</a></td><td width="40%" align="right" valign="top">
</td></tr></table></div></body></html>
diff --git a/help/C/html/ch05s07.html b/help/C/html/ch05s07.html
index 9b75200..3a6ddbd 100644
--- a/help/C/html/ch05s07.html
+++ b/help/C/html/ch05s07.html
@@ -63,10 +63,12 @@ returns 3.
Element by element back division.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>
The mod operator. This does not turn on the <a class="link" href="ch05s06.html" title="Modular
Evaluation">modular mode</a>, but
- just returns the remainder of <strong class="userinput"><code>a/b</code></strong>.
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
- Element by element the mod operator. Returns the remainder
- after element by element integer <strong class="userinput"><code>a./b</code></strong>.
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
</p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>
Modular evaluation operator. The expression <code class="varname">a</code>
is evaluated modulo <code class="varname">b</code>. See <a class="xref" href="ch05s06.html"
title="Modular Evaluation">the section called “Modular Evaluation”</a>.
@@ -102,21 +104,21 @@ returns 3.
greater than or equal to
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
- (can also be combine with the greater than operator).
+ (and they can also be combined with the greater than operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
Less than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
less than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
- (can also be combine with the less than or equal to operator).
+ (they can also be combined with the less than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
Greater than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
greater than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
- (can also be combine with the greater than or equal to operator).
+ (they can also be combined with the greater than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>
Comparison operator. If <code class="varname">a</code> is equal to
<code class="varname">b</code> it returns 0, if <code class="varname">a</code> is less
@@ -136,12 +138,12 @@ returns 3.
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
Logical xor.
- Returns true exactly one of
+ Returns true if exactly one of
<code class="varname">a</code> or <code class="varname">b</code> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
- Logical not. Returns the logical negation of <code class="varname">a</code>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>
@@ -160,7 +162,7 @@ returns 3.
Get element of a matrix in row <code class="varname">b</code> and column
<code class="varname">c</code>. If <code class="varname">b</code>,
<code class="varname">c</code> are vectors, then this gets the corresponding
- rows columns or submatrices.
+ rows, columns or submatrices.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>
Get row of a matrix (or multiple rows if <code class="varname">b</code> is a vector).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>
@@ -207,8 +209,8 @@ returns 3.
point numbers and is ever so slightly more precise than
<strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
- Make a imaginary number (multiply <code class="varname">a</code> by the
- imaginary). Note that normally the number <code class="varname">i</code> is
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
</p><pre class="programlisting">(a)*1i
</pre><p>
diff --git a/help/C/html/ch06s05.html b/help/C/html/ch06s05.html
index 9a9754d..3e2997e 100644
--- a/help/C/html/ch06s05.html
+++ b/help/C/html/ch06s05.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Global Variables and
Scope of Variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch06.html" title="Chapter 6. Programming with
GEL"><link rel="prev" href="ch06s04.html" title="Comparison Operators"><link rel="next" href="ch06s06.html"
title="Parameter variables"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Global Variables and Scope of Variables</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch06s04.html">Prev</a> </td><th width="60%" align="center">Chapter 6. Programming with
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Next</a></td></tr></table><hr></div><div class="sect1"><div cl
ass="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Global Variables and Scope of Variables</h2></div></div></div><p>
GEL is a
- <a class="ulink" href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
dynamically scoped language</a>. We will explain what this
means below. That is, normal variables and functions are dynamically
scoped. The exception are
diff --git a/help/C/html/ch07s02.html b/help/C/html/ch07s02.html
index ea65d04..2e4bf11 100644
--- a/help/C/html/ch07s02.html
+++ b/help/C/html/ch07s02.html
@@ -3,10 +3,32 @@
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
- as just sequence of lines as if were entered on the command line.
+ as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
- parentheses).
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
</p><p>
The following code will produce an error when entered on the top
level of a program, while it will work just fine in a function.
diff --git a/help/C/html/ch11s04.html b/help/C/html/ch11s04.html
index af54505..9caf8ca 100644
--- a/help/C/html/ch11s04.html
+++ b/help/C/html/ch11s04.html
@@ -2,26 +2,26 @@
Catalan's Constant, approximately 0.915... It is defined to be the series where terms are
<strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, where <code class="varname">k</code>
ranges from 0 to infinity.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Aliases: <code class="function">gamma</code></p><p>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>The
Golden Ratio.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
round and uniform.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
is the exponential function
@@ -30,7 +30,7 @@
several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>
@@ -38,7 +38,7 @@
to its diameter. This is approximately 3.14159265359...
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Parameters </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Numeric</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s05.html b/help/C/html/ch11s05.html
index b05e4c3..1519794 100644
--- a/help/C/html/ch11s05.html
+++ b/help/C/html/ch11s05.html
@@ -5,7 +5,7 @@
to <strong class="userinput"><code>|x|</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
<a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
<a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
@@ -14,16 +14,16 @@ for more information.
</p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Replace very small number with zero.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Aliases: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calculates the complex conjugate of the complex number <code
class="varname">z</code>. If <code class="varname">z</code> is a vector or matrix,
all its elements are conjugated.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Get the denominator of a rational number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Return the fractional part of a number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Aliases: <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Division without remainder.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
<strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
<strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Check if argument is a possibly complex rational number.
That is, if both real and imaginary parts are
@@ -32,10 +32,10 @@ all its elements are conjugated.</p><p>
are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Check if argument is an integer (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Aliases: <code
class="function">IsNaturalNumber</code></p><p>Check if argument is a positive real integer. Note that
we accept the convention that 0 is not a natural number.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Check if argument is a rational number (non-complex). Of course rational simply means "not
stored as a floating point number."</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (num)</pre><p>Check if
argument is a real number.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Get
the numerator of a rational number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Aliases: <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Aliases: <code class="function">sign</code></p><p>Return the sign of a
number. That is returns
<code class="literal">-1</code> if value is negative,
<code class="literal">0</code> if value is zero and
@@ -61,12 +61,12 @@ value then <code class="function">Sign</code> returns the direction or 0.
logarithm</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Make number a floating point value. That is returns the floating point
representation of the number <code class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Aliases: <code
class="function">Floor</code></p><p>Get the highest integer less than or equal to <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>The natural logarithm, the logarithm to base <code
class="varname">e</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logarithm of <code
class="varname">x</code> base <code class="varname">b</code> (calls <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> if in modulo
mode), if base is not given, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> is used.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logarithm of <code
class="varname">x</code> base 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Aliases: <code
class="function">lg</code></p><p>Logarithm of <code class="varname">x</code> base 2.</p></dd><dt><span
class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synop
sis">max (a,args...)</pre><p>Aliases: <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Returns the maximum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,args...)</pre><p>Aliases: <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Returns the minimum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(size...)</pre><p>Generate random float in the range <code class="literal">[0,1)</code>.
diff --git a/help/C/html/ch11s06.html b/help/C/html/ch11s06.html
index 71fe61e..fc255c7 100644
--- a/help/C/html/ch11s06.html
+++ b/help/C/html/ch11s06.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometry</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s05.html" title="Numeric"><link
rel="next" href="ch11s07.html" title="Number Theory"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometry</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a na
me="genius-gel-function-list-trigonometry"></a>Trigonometry</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Aliases: <code
class="function">arccos</code></p><p>The arccos (inverse cos) function.</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Aliases: <code
class="function">arccosh</code></p><p>The arccosh (inverse cosh) function.</p></dd><dt><span class="term"><a
name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Aliases: <code
class="function">arccot</code></p><p>The arccot (inverse cot) function.</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Aliases: <code
class="function">arccoth</code></p><p>The arccoth (inverse coth) function.</p></dd><dt><spa
n class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc
(x)</pre><p>Aliases: <code class="function">arccsc</code></p><p>The inverse cosecant
function.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Aliases: <code class="function">arccsch</code></p><p>The inverse
hyperbolic cosecant function.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Aliases: <code
class="function">arcsec</code></p><p>The inverse secant function.</p></dd><dt><span class="term"><a
name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech (x)</pre><p>Aliases: <code
class="function">arcsech</code></p><p>The inverse hyperbolic secant function.</p></dd><dt><span
class="term"><a name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin
(x)</pre><p>Aliases: <code class="function">arcsi
n</code></p><p>The arcsin (inverse sin) function.</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Aliases: <code
class="function">arcsinh</code></p><p>The arcsinh (inverse sinh) function.</p></dd><dt><span class="term"><a
name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Aliases: <code
class="function">arctan</code></p><p>Calculates the arctan (inverse tan) function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Aliases: <code class="function">arctanh</code></p><p>The arctanh (inverse
tanh) function.</p></dd><dt><span class="term"><a name="gel-function-atan2"></a>atan2</span></dt><dd><pre
class="synopsis">atan2 (y, x)</pre><p>Aliases: <code class="function">arctan2</code></p><p>Calculates the
arctan2 function. If
<strong class="userinput"><code>x>0</code></strong> then it returns
@@ -11,11 +11,11 @@
rather than failing.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Calculates the cosine function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Calculates the hyperbolic cosine function.</p><p>
See
@@ -23,7 +23,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>The cotangent function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>The hyperbolic cotangent function.</p><p>
See
@@ -31,7 +31,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>The cosecant function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>The hyperbolic cosecant function.</p><p>
See
@@ -39,7 +39,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>The secant function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>The hyperbolic secant function.</p><p>
See
@@ -47,7 +47,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Calculates the sine function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Calculates the hyperbolic sine function.</p><p>
See
@@ -55,7 +55,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calculates the tan function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>The hyperbolic tangent function.</p><p>
See
diff --git a/help/C/html/ch11s07.html b/help/C/html/ch11s07.html
index fada9af..67c9721 100644
--- a/help/C/html/ch11s07.html
+++ b/help/C/html/ch11s07.html
@@ -8,14 +8,14 @@
<a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Return the <code class="varname">n</code>th Bernoulli number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Aliases: <code class="function">CRT</code></p><p>Find the
<code class="varname">x</code> that solves the system given by
the vector <code class="varname">a</code> and modulo the elements of
<code class="varname">m</code>, using the Chinese Remainder Theorem.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Given two factorizations, give the factorization of the
@@ -23,7 +23,7 @@
F<sub>q</sub>, the finite field of order <code class="varname">q</code>, where <code
class="varname">q</code>
is a prime, using the Silver-Pohlig-Hellman algorithm.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Checks divisibility (if <code class="varname">m</code> divides <code
class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>
@@ -32,7 +32,7 @@
relatively prime to <code class="varname">n</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>
@@ -52,7 +52,7 @@
1 2 1]</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>
Return all factors of <code class="varname">n</code> in a vector. This
includes all the non-prime factors as well. It includes 1 and the
@@ -75,7 +75,7 @@
of two factors that are very close to each other.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Find the first primitive element in F<sub>q</sub>, the
finite
group of order <code class="varname">q</code>. Of course <code class="varname">q</code> must be a
prime.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Find a random primitive element in F<sub>q</sub>,
the finite
group of order <code class="varname">q</code> (q must be a prime).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of n in F<sub>q</sub>, the finite
@@ -99,7 +99,7 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -112,8 +112,8 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.
</p></dd><dt><span class="term"><a name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre
class="synopsis">IsOdd (n)</pre><p>Tests if an integer is odd.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
Check an integer for being a perfect square of an integer. The number must
- be a real integer. Negative integers are of course never perfect
- squares of real integers.
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
</p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>
Tests primality of integers, for numbers less than 2.5e10 the
answer is deterministic (if Riemann hypothesis is true). For
@@ -151,12 +151,12 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returns the <code class="varname">n</code>th Lucas number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Return all maximal prime power factors of a
number.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>
@@ -170,7 +170,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -185,7 +185,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
better on smaller integers.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
@@ -194,7 +194,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
result is deterministic.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Returns inverse of n mod m.</p><p>
diff --git a/help/C/html/ch11s08.html b/help/C/html/ch11s08.html
index 3ccdc54..17ac9e5 100644
--- a/help/C/html/ch11s08.html
+++ b/help/C/html/ch11s08.html
@@ -1,11 +1,11 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Matrix
Manipulation</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL
functions"><link rel="prev" href="ch11s07.html" title="Number Theory"><link rel="next" href="ch11s09.html"
title="Linear Algebra"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Matrix Manipulation</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s07.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s09.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" styl
e="clear: both"><a name="genius-gel-function-list-matrix"></a>Matrix Manipulation</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Apply a function over all entries of a matrix and return a matrix of the
results.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Apply a function over all entries of 2 matrices (or 1
value and 1 matrix) and return a matrix of the results.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Gets
the columns of a matrix as a horizontal vector.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre class="synopsis">
ComplementSubmatrix (m,r,c)</pre><p>Remove column(s) and row(s) from a matrix.</p></dd><dt><span
class="term"><a name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre
class="synopsis">CompoundMatrix (k,A)</pre><p>Calculate the kth compound matrix of A.</p></dd><dt><span
class="term"><a name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
- Count the number of zero columns in a matrix. For example
- once your column reduce a matrix you can use this to find
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
</p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Delete a column of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Delete a row of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Gets the diagonal entries of a matrix as a column vector.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
See
@@ -28,7 +28,7 @@
<strong class="userinput"><code>5</code></strong>, we return <strong
class="userinput"><code>[1,4,5]</code></strong>. If
<code class="varname">msize</code> is 0, we always return <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Is
a matrix diagonal.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Check if a matrix is the identity matrix. Automatically returns <code
class="constant">false</code>
if the matrix is not square. Also works on numbers, in which
@@ -37,12 +37,12 @@
no error is generated and <code class="constant">false</code> is returned.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Is a matrix lower triangular. That is, are all the entries
above the diagonal zero.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Check if a matrix is non-negative, that is if each element
is non-negative.
Do not confuse positive matrices with positive semi-definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Check if a matrix is positive, that is if each element is
positive (and hence real). In particular, no element is 0. Do not confuse
positive matrices with positive definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Check if a matrix is a matrix of rational (non-complex)
numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Check if a matrix is a matrix of real (non-complex) numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>
Check if a matrix is square, that is its width is equal to
@@ -62,7 +62,7 @@ functions make this check. Values can be any number including complex numbers.<
<strong class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> is the same as
<strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Make column vector out of matrix by putting columns above
each other. Returns <code class="constant">null</code> when given <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>
diff --git a/help/C/html/ch11s09.html b/help/C/html/ch11s09.html
index d405579..0864171 100644
--- a/help/C/html/ch11s09.html
+++ b/help/C/html/ch11s09.html
@@ -50,7 +50,7 @@ result as a vector and not added together.</p></dd><dt><span class="term"><a nam
diagonal).
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Get the eigenvectors of a square matrix. Optionally get also
@@ -58,7 +58,7 @@ the eigenvalues and their algebraic multiplicities.
Currently only works for matrices of size up to 2 by 2.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Apply the Gram-Schmidt process (to the columns) with respect to
@@ -152,7 +152,7 @@ determinant.
of two matrices.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
@@ -182,7 +182,7 @@ determinant.
and <code class="varname">U</code> to <code class="constant">null</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Get the <code class="varname">i</code>-<code class="varname">j</code>
minor of a matrix.</p><p>
@@ -218,7 +218,7 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<code class="varname">Q</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Return the Rayleigh quotient (also called the Rayleigh-Ritz
quotient or ratio) of a matrix and a vector.</p><p>
@@ -241,45 +241,45 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</p></dd><dt><span
class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Return the matrix corresponding
to rotation around origin in R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
x-axis.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (angle)</pre><p>Return the matrix corresponding to rotation around origin in
R<sup>3</sup> about the
y-axis.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
z-axis.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Get a basis matrix for the rowspace of a matrix.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Evaluate (v,w) with respect to the sesquilinear form given
by the matrix A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Return a function that evaluates two vectors with
respect to the sesquilinear form given by A.</p></dd><dt><span class="term"><a name="gel
-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returns the Smith normal form of a matrix over fields (will
end up with 1's on the diagonal).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Solve linear system Mx=V, return solution V if there
is a unique solution, <code class="constant">null</code> otherwise. Extra two reference parameters can
optionally be used to get the reduced M and V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Return the Toeplitz matrix constructed given the first column c
and (optionally) the first row r. If only the column c is given then it is
conjugated and the nonconjugated version is used for the first row to give a
Hermitian matrix (if the first element is real of course).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Aliases: <code class="function">trace</code></p><p>Calculate the trace of
a matrix. That is the sum of the diagonal elements.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Transpose of a matrix. This is the same as the
<strong class="userinput"><code>.'</code></strong> operator.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Aliases: <code class="function">vander</code></p><p>Return the
Vandermonde matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>The angle of two vectors with respect to inner product given by
<code class="varname">B</code>. If <code class="varname">B</code> is not given then the standard
Hermitian product is used. <code class="varname">B</code> can either be a sesquilinear
function of two arguments or it can be a matrix giving a sesquilinear form.
</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>The direct sum of the vector spaces M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersection of the subspaces given by M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>The sum of the vector spaces M and N, that is {w | w=m+n, m
in M, n in N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Aliases: <code class="function">Adjugate</code></p><p>Get the classical
adjoint (adjugate) of a matrix.</p></dd><dt><span class="term"><a name="gel-function-cref"></a>cref</spa
n></dt><dd><pre class="synopsis">cref (M)</pre><p>Aliases: <code class="function">CREF</code> <code
class="function">ColumnReducedEchelonForm</code></p><p>Compute the Column Reduced Echelon
Form.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Aliases: <code class="function">Determinant</code></p><p>Get the determinant
of a matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Aliases: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Get the row echelon form of a matrix. That is, apply gaussian
elimination but not backaddition to <code class="varname">M</code>. The pivot rows are
divided to make all pivots 1.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Aliases: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Get the reduced row echelon form of a matrix. That is,
apply gaussian elimination together with backaddition to <code class="varname">M</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s08.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s10.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Matrix Manipulation
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Combinatorics</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s10.html b/help/C/html/ch11s10.html
index f41e157..24c2b71 100644
--- a/help/C/html/ch11s10.html
+++ b/help/C/html/ch11s10.html
@@ -3,7 +3,10 @@
<a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Get all combinations of k numbers from 1 to n as a vector of vectors.
(See also <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)
-</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
See
<a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Factorial: <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
@@ -20,17 +23,18 @@
<strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
- Calculate the Frobenius number. That is calculate smallest
+ Calculate the Frobenius number. That is calculate largest
number that cannot be given as a non-negative integer linear
combination of a given vector of non-negative integers.
The vector can be given as separate numbers or a single vector.
All the numbers given should have GCD of 1.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(combining_rule)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>
Find the vector <code class="varname">c</code> of non-negative integers
@@ -40,8 +44,18 @@
of non-negative integers.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coeffi
cients. Takes a vector of
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coefficients. Takes a vector of
<code class="varname">k</code>
non-negative integers and computes the multinomial coefficient.
This corresponds to the coefficient in the homogeneous polynomial
@@ -57,7 +71,7 @@
<strong class="userinput"><code>Binomial(a+b,b)</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Get combination that would come after v in call to
@@ -77,6 +91,9 @@ do (
) while not IsNull(n:=NextCombination(n,6));</code></strong>
</pre><p>
See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Get the Pascal's triangle as a matrix. This will return
an <code class="varname">i</code>+1 by <code class="varname">i</code>+1 lower diagonal
matrix that is the Pascal's triangle after <code class="varname">i</code>
@@ -86,7 +103,7 @@ do (
</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Get all permutations of <code class="varname">k</code> numbers from 1 to <code
class="varname">n</code> as a vector of vectors.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Aliases: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k =
n(n+1)...(n+(k-1)).</p><p>
See
<a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
@@ -109,5 +126,5 @@ do (
<code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s11.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Linear Algebra </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Calculus</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s11.html b/help/C/html/ch11s11.html
index d38c659..d2ec9ea 100644
--- a/help/C/html/ch11s11.html
+++ b/help/C/html/ch11s11.html
@@ -25,7 +25,7 @@ the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, whil
<strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Either <code class="varname">a</code>
or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Try to calculate an infinite product for a single parameter
function.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Try to calculate an infinite product for a
double parameter function with func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Try to calculate an infinite sum for a single parameter function.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,inc)</pre><p>Try to calculate an infinite sum for a double
parameter function with func(arg,n).</p></dd><d
t><span class="term"><a name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre
class="synopsis">IsContinuous (f,x0)</pre><p>Try and see if a real-valued function is continuous at x0 by
calculating the limit there.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Test for differentiability by approximating the left and
right limits and comparing.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calculate the left limit of a real-valued function at x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Calculate the
limit of a real-valued function at x0. Tries to calculate both left and right limits.</p></dd><dt><span
class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</sp
an></dt><dd><pre class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integration by midpoint
rule.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Aliases: <code
class="function">NDerivative</code></p><p>Attempt to calculate numerical derivative.</p><p>
See
@@ -40,7 +40,7 @@ up to <code class="varname">N</code>th harmonic computed numerically. The coeff
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
@@ -50,7 +50,7 @@ trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the cosine Fourier series of
@@ -65,7 +65,7 @@ Note that <strong class="userinput"><code>a@(1)</code></strong> is
the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -76,7 +76,7 @@ only has cosine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the sine Fourier series of
@@ -88,7 +88,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -99,7 +99,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration by rule set in NumericalIntegralFunction of f
from a to b using NumericalIntegralSteps steps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Attempt to calculate numerical left
derivative.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Attempt to
calculate the limit of f(step_fun(i)) as i goes from 1 to N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">Nume
ricalRightDerivative (f,x0)</pre><p>Attempt to calculate numerical right derivative.</p></dd><dt><span
class="term"><a name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
<code class="function">f</code> with half period <code class="varname">L</code>. That
diff --git a/help/C/html/ch11s12.html b/help/C/html/ch11s12.html
index 6f38ad5..df529b3 100644
--- a/help/C/html/ch11s12.html
+++ b/help/C/html/ch11s12.html
@@ -1,21 +1,21 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Functions</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Genius
Manual"><link rel="up" href="ch11.html" title="Chapter 11. List of GEL functions"><link rel="prev"
href="ch11s11.html" title="Calculus"><link rel="next" href="ch11s13.html" title="Equation
Solving"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Functions</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL functions</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s13.html">Next</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name
="genius-gel-function-list-functions"></a>Functions</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Aliases: <code class="function">Arg</code> <code
class="function">arg</code></p><p>argument (angle) of complex number.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0 (x)</pre><p>Bessel
function of the first kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Bessel
function of the first kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Bessel
function of the first kind of order <code class="varname">n</code>. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Bessel
function of the second kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Bessel
function of the second kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Bessel
function of the second kind of order <code class="varname">n</code>. Only implemented for real
numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Aliases: <code class="function">erf</code></p><p>The error function, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
@@ -27,7 +27,7 @@
</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Aliases: <code class="function">Gamma</code></p><p>The Gamma function. Currently only
implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returns 1 if and only if all elements are equal.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>
The principal branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>.
@@ -38,7 +38,7 @@
See <a class="link" href="ch11s12.html#gel-function-LambertWm1"><code
class="function">LambertWm1</code></a> for the other real branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>
The minus-one branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>
@@ -48,29 +48,34 @@
See <a class="link" href="ch11s12.html#gel-function-LambertW"><code
class="function">LambertW</code></a> for the principal branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Find the first value where f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Moebius mapping of the disk to itself mapping a to 0.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity
respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Moebius mapping using the cross ratio taking
infinity to infinity and z2,z3 to 1 and 0 respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 1 and z3,z4 to 0 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 0 and z2,z4 to 1 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poisson kernel on D(0,R) (not normalized to
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Aliases: <code class="function">zeta</code></p><p>The Riemann zeta
function. Currently only implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>The unit step function is 0 for x<0, 1 otherwise. This is the
integral of the Dirac Delta function. Also called the Heaviside function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>
The <code class="function">cis</code> function, that is the same as
<strong class="userinput"><code>cos(x)+1i*sin(x)</code></strong>
@@ -78,5 +83,5 @@
<strong class="userinput"><code>sin(x)/x</code></strong>.
If you want the normalized function call <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
</p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Calculus </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Equation Solving</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s13.html b/help/C/html/ch11s13.html
index a383fdc..b9104d1 100644
--- a/help/C/html/ch11s13.html
+++ b/help/C/html/ch11s13.html
@@ -10,7 +10,7 @@
See
<a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
@@ -29,12 +29,12 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
@@ -73,7 +73,7 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Find root of a function using the bisection method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
@@ -102,7 +102,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Find zeros using Newton's method. <code class="varname">f</code> is
the function and <code class="varname">df</code> is the derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
@@ -116,7 +116,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>
Compute roots of a polynomial (degrees 1 through 4)
using one of the formulas for such polynomials.
@@ -139,8 +139,9 @@
Returns a column vector of the two solutions.
</p><p>
See
- <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>
Compute roots of a quartic (degree 4) polynomial using the
quartic formula. The polynomial should be given as a
@@ -152,7 +153,7 @@
See
<a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
@@ -168,14 +169,14 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
@@ -209,5 +210,5 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Functions </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Statistics</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s14.html b/help/C/html/ch11s14.html
index 00093d4..fe25d22 100644
--- a/help/C/html/ch11s14.html
+++ b/help/C/html/ch11s14.html
@@ -1,20 +1,27 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistics</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s13.html" title="Equation Solving"><link
rel="next" href="ch11s15.html" title="Polynomials"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Statistics</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Statistics</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average of an entire matrix.</p><p>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistics</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s13.html" title="Equation Solving"><link
rel="next" href="ch11s15.html" title="Polynomials"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Statistics</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Prev</a> </td><th width="60%" align="center">Chapter 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Statistics</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral of the GaussFunction from 0 to <code
class="varname">x</code> (area under the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>The normalized Gauss distribution function (the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Aliases: <code class="function">median</code></p><p>Calculate median of
an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calculate median of each row in a matrix and return a column
vector of the medians.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdevp</code></p><p>Calculate the population standard deviations of rows of a matrix and
return a vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdev</code></p><p>Calculate the standard deviations of rows of a matrix and return a
vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Aliases: <code class="function">stdev</code></p><p>Calculate
the standard deviation of a whole matrix.</p></dd></dl></div></div><div class="navfooter"><hr><table widt
h="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Equation Solving </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Polynomials</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s15.html b/help/C/html/ch11s15.html
index c4e84a1..2d0fc7d 100644
--- a/help/C/html/ch11s15.html
+++ b/help/C/html/ch11s15.html
@@ -17,5 +17,5 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Take second polynomial (as vector)
derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Take polynomial (as vector) derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Make function out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Make string out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Subtract two polynomials (as vectors).
</p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Trim zeros from a polynomial (as vector).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Statistics </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right"
valign="top"> Set Theory</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s18.html b/help/C/html/ch11s18.html
index e9d0a65..b2880c4 100644
--- a/help/C/html/ch11s18.html
+++ b/help/C/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscellaneous</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s17.html" title="Commutative
Algebra"><link rel="next" href="ch11s19.html" title="Symbolic Operations"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Miscellaneous</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Prev</a> </td><th width="60%" align="center">Chapter 11.
List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style
="clear: both"><a name="genius-gel-function-list-miscellaneous"></a>Miscellaneous</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a vector of ASCII values.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string t
o a vector of 0-based alphabet values (positions in the alphabet string), -1's for unknown
letters.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Commutative Algebra
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Symbolic Operations</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscellaneous</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius Manual"><link rel="up" href="ch11.html"
title="Chapter 11. List of GEL functions"><link rel="prev" href="ch11s17.html" title="Commutative
Algebra"><link rel="next" href="ch11s19.html" title="Symbolic Operations"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Miscellaneous</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Prev</a> </td><th width="60%" align="center">Chapter 11.
List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Next</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style
="clear: both"><a name="genius-gel-function-list-miscellaneous"></a>Miscellaneous</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Prev</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Commutative Algebra
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%"
align="right" valign="top"> Symbolic Operations</td></tr></table></div></body></html>
diff --git a/help/C/html/ch11s20.html b/help/C/html/ch11s20.html
index dc4e50f..b43e1c3 100644
--- a/help/C/html/ch11s20.html
+++ b/help/C/html/ch11s20.html
@@ -102,7 +102,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
</pre><p>
@@ -153,7 +153,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
</pre><p>
@@ -330,7 +330,7 @@ limits as <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.
<code class="varname">n</code> by 3 matrix for a longer polyline.
</p><p>
Extra parameters can be added to specify line color, thickness,
- arrows, the plotting window, or legend.
+ the plotting window, or legend.
You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
<strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>,
diff --git a/help/C/html/index.html b/help/C/html/index.html
index e1bf261..9ce6265 100644
--- a/help/C/html/index.html
+++ b/help/C/html/index.html
@@ -58,7 +58,7 @@
EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED OF
THE POSSIBILITY OF SUCH DAMAGES.
</p></li></ol></div><p>
- </p></div></div><div><div class="legalnotice"><a name="idm45453684982368"></a><p
class="legalnotice-title"><b>Feedback</b></p><p>
+ </p></div></div><div><div class="legalnotice"><a name="idm48"></a><p
class="legalnotice-title"><b>Feedback</b></p><p>
To report a bug or make a suggestion regarding the <span class="application">Genius Mathematics
Tool</span>
application or this manual, please visit the
<a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius
diff --git a/help/C/index.html b/help/C/index.html
new file mode 100644
index 0000000..9ce6265
--- /dev/null
+++ b/help/C/index.html
@@ -0,0 +1,71 @@
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Genius
Manual</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Manual for the Genius Math Tool."><link rel="home" href="index.html" title="Genius Manual"><link
rel="next" href="ch01.html" title="Chapter 1. Introduction"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Genius Manual</th></tr><tr><td width="20%" align="left"> </td><th
width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Next</a></td></tr></table><hr></div><div lang="en" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Genius Manual</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><span class="firstname">Jiří</sp
an> <span class="surname">Lebl</span></h3><div class="affiliation"><span class="orgname">Oklahoma State
University<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code> </p></div></div></div><div class="author"><h3 class="author"><span
class="firstname">Kai</span> <span class="surname">Willadsen</span></h3><div class="affiliation"><span
class="orgname">University of Queensland, Australia<br></span><div class="address"><p> <code
class="email"><<a class="email" href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">This manual describes version 1.0.22 of Genius.
+ </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><div class="legalnotice"><a
name="legalnotice"></a><p>
+ Permission is granted to copy, distribute and/or modify this
+ document under the terms of the GNU Free Documentation
+ License (GFDL), Version 1.1 or any later version published
+ by the Free Software Foundation with no Invariant Sections,
+ no Front-Cover Texts, and no Back-Cover Texts. You can find
+ a copy of the GFDL at this <a class="ulink" href="ghelp:fdl" target="_top">link</a> or in the file
COPYING-DOCS
+ distributed with this manual.
+ </p><p> This manual is part of a collection of GNOME manuals
+ distributed under the GFDL. If you want to distribute this
+ manual separately from the collection, you can do so by
+ adding a copy of the license to the manual, as described in
+ section 6 of the license.
+ </p><p>
+ Many of the names used by companies to distinguish their
+ products and services are claimed as trademarks. Where those
+ names appear in any GNOME documentation, and the members of
+ the GNOME Documentation Project are made aware of those
+ trademarks, then the names are in capital letters or initial
+ capital letters.
+ </p><p>
+ DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED
+ UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE
+ WITH THE FURTHER UNDERSTANDING THAT:
+
+ </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>DOCUMENT IS
PROVIDED ON AN "AS IS" BASIS,
+ WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR
+ IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES
+ THAT THE DOCUMENT OR MODIFIED VERSION OF THE
+ DOCUMENT IS FREE OF DEFECTS MERCHANTABLE, FIT FOR
+ A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE
+ RISK AS TO THE QUALITY, ACCURACY, AND PERFORMANCE
+ OF THE DOCUMENT OR MODIFIED VERSION OF THE
+ DOCUMENT IS WITH YOU. SHOULD ANY DOCUMENT OR
+ MODIFIED VERSION PROVE DEFECTIVE IN ANY RESPECT,
+ YOU (NOT THE INITIAL WRITER, AUTHOR OR ANY
+ CONTRIBUTOR) ASSUME THE COST OF ANY NECESSARY
+ SERVICING, REPAIR OR CORRECTION. THIS DISCLAIMER
+ OF WARRANTY CONSTITUTES AN ESSENTIAL PART OF THIS
+ LICENSE. NO USE OF ANY DOCUMENT OR MODIFIED
+ VERSION OF THE DOCUMENT IS AUTHORIZED HEREUNDER
+ EXCEPT UNDER THIS DISCLAIMER; AND
+ </p></li><li class="listitem"><p>UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL
+ THEORY, WHETHER IN TORT (INCLUDING NEGLIGENCE),
+ CONTRACT, OR OTHERWISE, SHALL THE AUTHOR,
+ INITIAL WRITER, ANY CONTRIBUTOR, OR ANY
+ DISTRIBUTOR OF THE DOCUMENT OR MODIFIED VERSION
+ OF THE DOCUMENT, OR ANY SUPPLIER OF ANY OF SUCH
+ PARTIES, BE LIABLE TO ANY PERSON FOR ANY
+ DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR
+ CONSEQUENTIAL DAMAGES OF ANY CHARACTER
+ INCLUDING, WITHOUT LIMITATION, DAMAGES FOR LOSS
+ OF GOODWILL, WORK STOPPAGE, COMPUTER FAILURE OR
+ MALFUNCTION, OR ANY AND ALL OTHER DAMAGES OR
+ LOSSES ARISING OUT OF OR RELATING TO USE OF THE
+ DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT,
+ EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED OF
+ THE POSSIBILITY OF SUCH DAMAGES.
+ </p></li></ol></div><p>
+ </p></div></div><div><div class="legalnotice"><a name="idm48"></a><p
class="legalnotice-title"><b>Feedback</b></p><p>
+ To report a bug or make a suggestion regarding the <span class="application">Genius Mathematics
Tool</span>
+ application or this manual, please visit the
+ <a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius
+ Web page</a>
+ or email me at <code class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z
com</a>></code>.
+ </p></div></div><div><div class="revhistory"><table style="border-style:solid; width:100%;"
summary="Revision History"><tr><th align="left" valign="top" colspan="2"><b>Revision
History</b></th></tr><tr><td align="left">Revision 0.2</td><td align="left">September 2016</td></tr><tr><td
align="left" colspan="2">
+ <p class="author">Jiri (George) Lebl
+ <code class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z
com</a>></code>
+ </p>
+ </td></tr></table></div></div><div><div class="abstract"><p
class="title"><b>Abstract</b></p><p>Manual for the Genius Math Tool.</p></div></div></div><hr></div><div
class="toc"><p><b>Table of Contents</b></p><dl class="toc"><dt><span class="chapter"><a href="ch01.html">1.
Introduction</a></span></dt><dt><span class="chapter"><a href="ch02.html">2. Getting
Started</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch02.html#genius-to-start">To Start <span
class="application">Genius Mathematics Tool</span></a></span></dt><dt><span class="sect1"><a
href="ch02s02.html">When You Start Genius</a></span></dt></dl></dd><dt><span class="chapter"><a
href="ch03.html">3. Basic Usage</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch03.html#genius-usage-workarea">Using the Work Area</a></span></dt><dt><span class="sect1"><a
href="ch03s02.html">To Create a New Program </a></span></dt><dt><span class="sect1"><a href="ch03s03.html">To
Open and Run a Program </a></span></dt></
dl></dd><dt><span class="chapter"><a href="ch04.html">4. Plotting</a></span></dt><dd><dl><dt><span
class="sect1"><a href="ch04.html#genius-line-plots">Line Plots</a></span></dt><dt><span class="sect1"><a
href="ch04s02.html">Parametric Plots</a></span></dt><dt><span class="sect1"><a href="ch04s03.html">Slopefield
Plots</a></span></dt><dt><span class="sect1"><a href="ch04s04.html">Vectorfield
Plots</a></span></dt><dt><span class="sect1"><a href="ch04s05.html">Surface
Plots</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch05.html">5. GEL
Basics</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch05.html#genius-gel-values">Values</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-numbers">Numbers</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-booleans">Booleans</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-strings">Strings</a></span></dt><dt><span class="sect2"><a href=
"ch05.html#genius-gel-values-null">Null</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s02.html">Using Variables</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-setting">Setting Variables</a></span></dt><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-built-in">Built-in Variables</a></span></dt><dt><span
class="sect2"><a href="ch05s02.html#genius-gel-previous-result">Previous Result
Variable</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch05s03.html">Using
Functions</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-defining">Defining Functions</a></span></dt><dt><span
class="sect2"><a href="ch05s03.html#genius-gel-functions-variable-argument-lists">Variable Argument
Lists</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-passing-functions">Passing Functions to
Functions</a></span></dt><dt><span class="sect2"><a href="c
h05s03.html#genius-gel-functions-operations">Operations on Functions</a></span></dt></dl></dd><dt><span
class="sect1"><a href="ch05s04.html">Separator</a></span></dt><dt><span class="sect1"><a
href="ch05s05.html">Comments</a></span></dt><dt><span class="sect1"><a href="ch05s06.html">Modular
Evaluation</a></span></dt><dt><span class="sect1"><a href="ch05s07.html">List of GEL
Operators</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch06.html">6. Programming with
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch06.html#genius-gel-conditionals">Conditionals</a></span></dt><dt><span class="sect1"><a
href="ch06s02.html">Loops</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-while">While Loops</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-for">For Loops</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-foreach">Foreach Loops</a></span></dt><dt><span class="sect
2"><a href="ch06s02.html#genius-gel-loops-break-continue">Break and
Continue</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch06s03.html">Sums and
Products</a></span></dt><dt><span class="sect1"><a href="ch06s04.html">Comparison
Operators</a></span></dt><dt><span class="sect1"><a href="ch06s05.html">Global Variables and Scope of
Variables</a></span></dt><dt><span class="sect1"><a href="ch06s06.html">Parameter
variables</a></span></dt><dt><span class="sect1"><a href="ch06s07.html">Returning</a></span></dt><dt><span
class="sect1"><a href="ch06s08.html">References</a></span></dt><dt><span class="sect1"><a
href="ch06s09.html">Lvalues</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch07.html">7.
Advanced Programming with GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch07.html#genius-gel-error-handling">Error Handling</a></span></dt><dt><span class="sect1"><a
href="ch07s02.html">Toplevel Syntax</a></span></dt><dt><span class="sect1"><a href="ch
07s03.html">Returning Functions</a></span></dt><dt><span class="sect1"><a href="ch07s04.html">True Local
Variables</a></span></dt><dt><span class="sect1"><a href="ch07s05.html">GEL Startup
Procedure</a></span></dt><dt><span class="sect1"><a href="ch07s06.html">Loading
Programs</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch08.html">8. Matrices in
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch08.html#genius-gel-matrix-support">Entering
Matrices</a></span></dt><dt><span class="sect1"><a href="ch08s02.html">Conjugate Transpose and Transpose
Operator</a></span></dt><dt><span class="sect1"><a href="ch08s03.html">Linear
Algebra</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch09.html">9. Polynomials in
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch09.html#genius-gel-polynomials-using">Using
Polynomials</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch10.html">10. Set Theory in
GEL</a></span></dt><dd><dl><dt><
span class="sect1"><a href="ch10.html#genius-gel-sets-using">Using Sets</a></span></dt></dl></dd><dt><span
class="chapter"><a href="ch11.html">11. List of GEL functions</a></span></dt><dd><dl><dt><span
class="sect1"><a href="ch11.html#genius-gel-function-list-commands">Commands</a></span></dt><dt><span
class="sect1"><a href="ch11s02.html">Basic</a></span></dt><dt><span class="sect1"><a
href="ch11s03.html">Parameters</a></span></dt><dt><span class="sect1"><a
href="ch11s04.html">Constants</a></span></dt><dt><span class="sect1"><a
href="ch11s05.html">Numeric</a></span></dt><dt><span class="sect1"><a
href="ch11s06.html">Trigonometry</a></span></dt><dt><span class="sect1"><a href="ch11s07.html">Number
Theory</a></span></dt><dt><span class="sect1"><a href="ch11s08.html">Matrix
Manipulation</a></span></dt><dt><span class="sect1"><a href="ch11s09.html">Linear
Algebra</a></span></dt><dt><span class="sect1"><a href="ch11s10.html">Combinatorics</a></span></dt><dt><span
class="sect1"><a
href="ch11s11.html">Calculus</a></span></dt><dt><span class="sect1"><a
href="ch11s12.html">Functions</a></span></dt><dt><span class="sect1"><a href="ch11s13.html">Equation
Solving</a></span></dt><dt><span class="sect1"><a href="ch11s14.html">Statistics</a></span></dt><dt><span
class="sect1"><a href="ch11s15.html">Polynomials</a></span></dt><dt><span class="sect1"><a
href="ch11s16.html">Set Theory</a></span></dt><dt><span class="sect1"><a href="ch11s17.html">Commutative
Algebra</a></span></dt><dt><span class="sect1"><a href="ch11s18.html">Miscellaneous</a></span></dt><dt><span
class="sect1"><a href="ch11s19.html">Symbolic Operations</a></span></dt><dt><span class="sect1"><a
href="ch11s20.html">Plotting</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch12.html">12.
Example Programs in GEL</a></span></dt><dt><span class="chapter"><a href="ch13.html">13.
Settings</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch13.html#genius-prefs-output">Output</a></span>
</dt><dt><span class="sect1"><a href="ch13s02.html">Precision</a></span></dt><dt><span class="sect1"><a
href="ch13s03.html">Terminal</a></span></dt><dt><span class="sect1"><a
href="ch13s04.html">Memory</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch14.html">14. About
<span class="application">Genius Mathematics Tool</span></a></span></dt></dl></div><div
class="list-of-figures"><p><b>List of Figures</b></p><dl><dt>2.1. <a href="ch02s02.html#mainwindow-fig"><span
class="application">Genius Mathematics Tool</span> Window</a></dt><dt>4.1. <a
href="ch04.html#lineplot-fig">Create Plot Window</a></dt><dt>4.2. <a href="ch04.html#lineplot2-fig">Plot
Window</a></dt><dt>4.3. <a href="ch04s02.html#paramplot-fig">Parametric Plot Tab</a></dt><dt>4.4. <a
href="ch04s02.html#paramplot2-fig">Parametric Plot</a></dt><dt>4.5. <a
href="ch04s05.html#surfaceplot-fig">Surface Plot</a></dt></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr>
<td width="40%" align="left"> </td><td width="20%" align="center"> </td><td width="40%" align="right"> <a
accesskey="n" href="ch01.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top"> </td><td
width="20%" align="center"> </td><td width="40%" align="right" valign="top"> Chapter 1.
Introduction</td></tr></table></div></body></html>
diff --git a/help/Makefile.am b/help/Makefile.am
index e056eda..2ad6d9d 100644
--- a/help/Makefile.am
+++ b/help/Makefile.am
@@ -1,24 +1,12 @@
+###################################################################
+# THIS FILE IS AUTOGENERATED DO NOT EDIT. EDIT make-makefile-am.sh
+###################################################################
-# After updating docs run ./update-xml-to-txt-html.sh
-# Read that script for what it does. I'll run this script before releasing.
-# The output is stored in git and it is not run on make or make dist or some
-# such.
-#
-# to update an existing po file, you can run ./update-po.sh LANG
-# or just use xml2po manually
-
-
-# FIXME: this should be some sort of a loop
-# DOC_LINGUAS = cs de el es fr pt_BR ru sv
-# If adding languages make sure to also update the update-xml-to-txt-html.sh
+#########################################################
#C
-THE_CFIGURES = C/figures/parametric.png \
- C/figures/genius_window.png \
- C/figures/line_plot_graph.png \
- C/figures/line_plot.png \
- C/figures/parametric_graph.png \
- C/figures/surface_graph.png
+
+THE_CFIGURES = C/figures/genius_window.png C/figures/line_plot_graph.png C/figures/line_plot.png
C/figures/parametric_graph.png C/figures/parametric.png C/figures/surface_graph.png
manualxmlCdir = $(datadir)/genius/help/C
manualxmlC_DATA = C/genius.xml C/legal.xml
@@ -26,17 +14,79 @@ manualxmlCfiguresdir = $(datadir)/genius/help/C/figures
manualxmlCfigures_DATA = $(THE_CFIGURES)
manualhtmlCdir = $(datadir)/genius/help/C/html
-manualhtmlC_DATA = C/html/*.html
+manualhtmlC_DATA = C/html/ch01.html \
+ C/html/ch02.html \
+ C/html/ch02s02.html \
+ C/html/ch03.html \
+ C/html/ch03s02.html \
+ C/html/ch03s03.html \
+ C/html/ch04.html \
+ C/html/ch04s02.html \
+ C/html/ch04s03.html \
+ C/html/ch04s04.html \
+ C/html/ch04s05.html \
+ C/html/ch05.html \
+ C/html/ch05s02.html \
+ C/html/ch05s03.html \
+ C/html/ch05s04.html \
+ C/html/ch05s05.html \
+ C/html/ch05s06.html \
+ C/html/ch05s07.html \
+ C/html/ch06.html \
+ C/html/ch06s02.html \
+ C/html/ch06s03.html \
+ C/html/ch06s04.html \
+ C/html/ch06s05.html \
+ C/html/ch06s06.html \
+ C/html/ch06s07.html \
+ C/html/ch06s08.html \
+ C/html/ch06s09.html \
+ C/html/ch07.html \
+ C/html/ch07s02.html \
+ C/html/ch07s03.html \
+ C/html/ch07s04.html \
+ C/html/ch07s05.html \
+ C/html/ch07s06.html \
+ C/html/ch08.html \
+ C/html/ch08s02.html \
+ C/html/ch08s03.html \
+ C/html/ch09.html \
+ C/html/ch10.html \
+ C/html/ch11.html \
+ C/html/ch11s02.html \
+ C/html/ch11s03.html \
+ C/html/ch11s04.html \
+ C/html/ch11s05.html \
+ C/html/ch11s06.html \
+ C/html/ch11s07.html \
+ C/html/ch11s08.html \
+ C/html/ch11s09.html \
+ C/html/ch11s10.html \
+ C/html/ch11s11.html \
+ C/html/ch11s12.html \
+ C/html/ch11s13.html \
+ C/html/ch11s14.html \
+ C/html/ch11s15.html \
+ C/html/ch11s16.html \
+ C/html/ch11s17.html \
+ C/html/ch11s18.html \
+ C/html/ch11s19.html \
+ C/html/ch11s20.html \
+ C/html/ch12.html \
+ C/html/ch13.html \
+ C/html/ch13s02.html \
+ C/html/ch13s03.html \
+ C/html/ch13s04.html \
+ C/html/ch14.html \
+ C/html/index.html
manualhtmlCfiguresdir = $(datadir)/genius/help/C/html/figures
manualhtmlCfigures_DATA = $(THE_CFIGURES)
+
+#########################################################
#cs
-THE_csFIGURES = cs/figures/parametric.png \
- cs/figures/genius_window.png \
- cs/figures/line_plot_graph.png \
- cs/figures/line_plot.png \
- cs/figures/parametric_graph.png \
- cs/figures/surface_graph.png
+
+THE_csFIGURES = cs/figures/genius_window.png cs/figures/line_plot_graph.png cs/figures/line_plot.png
cs/figures/parametric_graph.png cs/figures/parametric.png cs/figures/surface_graph.png
manualxmlcsdir = $(datadir)/genius/help/cs
manualxmlcs_DATA = cs/genius.xml
@@ -44,17 +94,79 @@ manualxmlcsfiguresdir = $(datadir)/genius/help/cs/figures
manualxmlcsfigures_DATA = $(THE_csFIGURES)
manualhtmlcsdir = $(datadir)/genius/help/cs/html
-manualhtmlcs_DATA = cs/html/*.html
+manualhtmlcs_DATA = cs/html/ch01.html \
+ cs/html/ch02.html \
+ cs/html/ch02s02.html \
+ cs/html/ch03.html \
+ cs/html/ch03s02.html \
+ cs/html/ch03s03.html \
+ cs/html/ch04.html \
+ cs/html/ch04s02.html \
+ cs/html/ch04s03.html \
+ cs/html/ch04s04.html \
+ cs/html/ch04s05.html \
+ cs/html/ch05.html \
+ cs/html/ch05s02.html \
+ cs/html/ch05s03.html \
+ cs/html/ch05s04.html \
+ cs/html/ch05s05.html \
+ cs/html/ch05s06.html \
+ cs/html/ch05s07.html \
+ cs/html/ch06.html \
+ cs/html/ch06s02.html \
+ cs/html/ch06s03.html \
+ cs/html/ch06s04.html \
+ cs/html/ch06s05.html \
+ cs/html/ch06s06.html \
+ cs/html/ch06s07.html \
+ cs/html/ch06s08.html \
+ cs/html/ch06s09.html \
+ cs/html/ch07.html \
+ cs/html/ch07s02.html \
+ cs/html/ch07s03.html \
+ cs/html/ch07s04.html \
+ cs/html/ch07s05.html \
+ cs/html/ch07s06.html \
+ cs/html/ch08.html \
+ cs/html/ch08s02.html \
+ cs/html/ch08s03.html \
+ cs/html/ch09.html \
+ cs/html/ch10.html \
+ cs/html/ch11.html \
+ cs/html/ch11s02.html \
+ cs/html/ch11s03.html \
+ cs/html/ch11s04.html \
+ cs/html/ch11s05.html \
+ cs/html/ch11s06.html \
+ cs/html/ch11s07.html \
+ cs/html/ch11s08.html \
+ cs/html/ch11s09.html \
+ cs/html/ch11s10.html \
+ cs/html/ch11s11.html \
+ cs/html/ch11s12.html \
+ cs/html/ch11s13.html \
+ cs/html/ch11s14.html \
+ cs/html/ch11s15.html \
+ cs/html/ch11s16.html \
+ cs/html/ch11s17.html \
+ cs/html/ch11s18.html \
+ cs/html/ch11s19.html \
+ cs/html/ch11s20.html \
+ cs/html/ch12.html \
+ cs/html/ch13.html \
+ cs/html/ch13s02.html \
+ cs/html/ch13s03.html \
+ cs/html/ch13s04.html \
+ cs/html/ch14.html \
+ cs/html/index.html
manualhtmlcsfiguresdir = $(datadir)/genius/help/cs/html/figures
manualhtmlcsfigures_DATA = $(THE_csFIGURES)
+
+#########################################################
#de
-THE_deFIGURES = de/figures/parametric.png \
- de/figures/genius_window.png \
- de/figures/line_plot_graph.png \
- de/figures/line_plot.png \
- de/figures/parametric_graph.png \
- de/figures/surface_graph.png
+
+THE_deFIGURES = de/figures/genius_window.png de/figures/line_plot_graph.png de/figures/line_plot.png
de/figures/parametric_graph.png de/figures/parametric.png de/figures/surface_graph.png
manualxmldedir = $(datadir)/genius/help/de
manualxmlde_DATA = de/genius.xml
@@ -62,39 +174,239 @@ manualxmldefiguresdir = $(datadir)/genius/help/de/figures
manualxmldefigures_DATA = $(THE_deFIGURES)
manualhtmldedir = $(datadir)/genius/help/de/html
-manualhtmlde_DATA = de/html/*.html
+manualhtmlde_DATA = de/html/ch01.html \
+ de/html/ch02.html \
+ de/html/ch02s02.html \
+ de/html/ch03.html \
+ de/html/ch03s02.html \
+ de/html/ch03s03.html \
+ de/html/ch04.html \
+ de/html/ch04s02.html \
+ de/html/ch04s03.html \
+ de/html/ch04s04.html \
+ de/html/ch04s05.html \
+ de/html/ch05.html \
+ de/html/ch05s02.html \
+ de/html/ch05s03.html \
+ de/html/ch05s04.html \
+ de/html/ch05s05.html \
+ de/html/ch05s06.html \
+ de/html/ch05s07.html \
+ de/html/ch06.html \
+ de/html/ch06s02.html \
+ de/html/ch06s03.html \
+ de/html/ch06s04.html \
+ de/html/ch06s05.html \
+ de/html/ch06s06.html \
+ de/html/ch06s07.html \
+ de/html/ch06s08.html \
+ de/html/ch06s09.html \
+ de/html/ch07.html \
+ de/html/ch07s02.html \
+ de/html/ch07s03.html \
+ de/html/ch07s04.html \
+ de/html/ch07s05.html \
+ de/html/ch07s06.html \
+ de/html/ch08.html \
+ de/html/ch08s02.html \
+ de/html/ch08s03.html \
+ de/html/ch09.html \
+ de/html/ch10.html \
+ de/html/ch11.html \
+ de/html/ch11s02.html \
+ de/html/ch11s03.html \
+ de/html/ch11s04.html \
+ de/html/ch11s05.html \
+ de/html/ch11s06.html \
+ de/html/ch11s07.html \
+ de/html/ch11s08.html \
+ de/html/ch11s09.html \
+ de/html/ch11s10.html \
+ de/html/ch11s11.html \
+ de/html/ch11s12.html \
+ de/html/ch11s13.html \
+ de/html/ch11s14.html \
+ de/html/ch11s15.html \
+ de/html/ch11s16.html \
+ de/html/ch11s17.html \
+ de/html/ch11s18.html \
+ de/html/ch11s19.html \
+ de/html/ch11s20.html \
+ de/html/ch12.html \
+ de/html/ch13.html \
+ de/html/ch13s02.html \
+ de/html/ch13s03.html \
+ de/html/ch13s04.html \
+ de/html/ch14.html \
+ de/html/index.html
manualhtmldefiguresdir = $(datadir)/genius/help/de/html/figures
manualhtmldefigures_DATA = $(THE_deFIGURES)
+
+#########################################################
#el
+
+THE_elFIGURES = C/figures/genius_window.png C/figures/line_plot_graph.png C/figures/line_plot.png
C/figures/parametric_graph.png C/figures/parametric.png C/figures/surface_graph.png
+
manualxmleldir = $(datadir)/genius/help/el
manualxmlel_DATA = el/genius.xml
manualxmlelfiguresdir = $(datadir)/genius/help/el/figures
-manualxmlelfigures_DATA = $(THE_CFIGURES)
+manualxmlelfigures_DATA = $(THE_elFIGURES)
manualhtmleldir = $(datadir)/genius/help/el/html
-manualhtmlel_DATA = el/html/*.html
+manualhtmlel_DATA = el/html/ch01.html \
+ el/html/ch02.html \
+ el/html/ch02s02.html \
+ el/html/ch03.html \
+ el/html/ch03s02.html \
+ el/html/ch03s03.html \
+ el/html/ch04.html \
+ el/html/ch04s02.html \
+ el/html/ch04s03.html \
+ el/html/ch04s04.html \
+ el/html/ch04s05.html \
+ el/html/ch05.html \
+ el/html/ch05s02.html \
+ el/html/ch05s03.html \
+ el/html/ch05s04.html \
+ el/html/ch05s05.html \
+ el/html/ch05s06.html \
+ el/html/ch05s07.html \
+ el/html/ch06.html \
+ el/html/ch06s02.html \
+ el/html/ch06s03.html \
+ el/html/ch06s04.html \
+ el/html/ch06s05.html \
+ el/html/ch06s06.html \
+ el/html/ch06s07.html \
+ el/html/ch06s08.html \
+ el/html/ch06s09.html \
+ el/html/ch07.html \
+ el/html/ch07s02.html \
+ el/html/ch07s03.html \
+ el/html/ch07s04.html \
+ el/html/ch07s05.html \
+ el/html/ch07s06.html \
+ el/html/ch08.html \
+ el/html/ch08s02.html \
+ el/html/ch08s03.html \
+ el/html/ch09.html \
+ el/html/ch10.html \
+ el/html/ch11.html \
+ el/html/ch11s02.html \
+ el/html/ch11s03.html \
+ el/html/ch11s04.html \
+ el/html/ch11s05.html \
+ el/html/ch11s06.html \
+ el/html/ch11s07.html \
+ el/html/ch11s08.html \
+ el/html/ch11s09.html \
+ el/html/ch11s10.html \
+ el/html/ch11s11.html \
+ el/html/ch11s12.html \
+ el/html/ch11s13.html \
+ el/html/ch11s14.html \
+ el/html/ch11s15.html \
+ el/html/ch11s16.html \
+ el/html/ch11s17.html \
+ el/html/ch11s18.html \
+ el/html/ch11s19.html \
+ el/html/ch11s20.html \
+ el/html/ch12.html \
+ el/html/ch13.html \
+ el/html/ch13s02.html \
+ el/html/ch13s03.html \
+ el/html/ch13s04.html \
+ el/html/ch14.html \
+ el/html/index.html
manualhtmlelfiguresdir = $(datadir)/genius/help/el/html/figures
-manualhtmlelfigures_DATA = $(THE_CFIGURES)
+manualhtmlelfigures_DATA = $(THE_elFIGURES)
+
+#########################################################
#es
+
+THE_esFIGURES = C/figures/genius_window.png C/figures/line_plot_graph.png C/figures/line_plot.png
C/figures/parametric_graph.png C/figures/parametric.png C/figures/surface_graph.png
+
manualxmlesdir = $(datadir)/genius/help/es
manualxmles_DATA = es/genius.xml
manualxmlesfiguresdir = $(datadir)/genius/help/es/figures
manualxmlesfigures_DATA = $(THE_esFIGURES)
manualhtmlesdir = $(datadir)/genius/help/es/html
-manualhtmles_DATA = es/html/*.html
+manualhtmles_DATA = es/html/ch01.html \
+ es/html/ch02.html \
+ es/html/ch02s02.html \
+ es/html/ch03.html \
+ es/html/ch03s02.html \
+ es/html/ch03s03.html \
+ es/html/ch04.html \
+ es/html/ch04s02.html \
+ es/html/ch04s03.html \
+ es/html/ch04s04.html \
+ es/html/ch04s05.html \
+ es/html/ch05.html \
+ es/html/ch05s02.html \
+ es/html/ch05s03.html \
+ es/html/ch05s04.html \
+ es/html/ch05s05.html \
+ es/html/ch05s06.html \
+ es/html/ch05s07.html \
+ es/html/ch06.html \
+ es/html/ch06s02.html \
+ es/html/ch06s03.html \
+ es/html/ch06s04.html \
+ es/html/ch06s05.html \
+ es/html/ch06s06.html \
+ es/html/ch06s07.html \
+ es/html/ch06s08.html \
+ es/html/ch06s09.html \
+ es/html/ch07.html \
+ es/html/ch07s02.html \
+ es/html/ch07s03.html \
+ es/html/ch07s04.html \
+ es/html/ch07s05.html \
+ es/html/ch07s06.html \
+ es/html/ch08.html \
+ es/html/ch08s02.html \
+ es/html/ch08s03.html \
+ es/html/ch09.html \
+ es/html/ch10.html \
+ es/html/ch11.html \
+ es/html/ch11s02.html \
+ es/html/ch11s03.html \
+ es/html/ch11s04.html \
+ es/html/ch11s05.html \
+ es/html/ch11s06.html \
+ es/html/ch11s07.html \
+ es/html/ch11s08.html \
+ es/html/ch11s09.html \
+ es/html/ch11s10.html \
+ es/html/ch11s11.html \
+ es/html/ch11s12.html \
+ es/html/ch11s13.html \
+ es/html/ch11s14.html \
+ es/html/ch11s15.html \
+ es/html/ch11s16.html \
+ es/html/ch11s17.html \
+ es/html/ch11s18.html \
+ es/html/ch11s19.html \
+ es/html/ch11s20.html \
+ es/html/ch12.html \
+ es/html/ch13.html \
+ es/html/ch13s02.html \
+ es/html/ch13s03.html \
+ es/html/ch13s04.html \
+ es/html/ch14.html \
+ es/html/index.html
manualhtmlesfiguresdir = $(datadir)/genius/help/es/html/figures
manualhtmlesfigures_DATA = $(THE_esFIGURES)
+
+#########################################################
#fr
-THE_frFIGURES = fr/figures/parametric.png \
- fr/figures/genius_window.png \
- fr/figures/line_plot_graph.png \
- fr/figures/line_plot.png \
- fr/figures/parametric_graph.png \
- fr/figures/surface_graph.png
+
+THE_frFIGURES = fr/figures/genius_window.png fr/figures/line_plot_graph.png fr/figures/line_plot.png
fr/figures/parametric_graph.png fr/figures/parametric.png fr/figures/surface_graph.png
manualxmlfrdir = $(datadir)/genius/help/fr
manualxmlfr_DATA = fr/genius.xml
@@ -102,39 +414,239 @@ manualxmlfrfiguresdir = $(datadir)/genius/help/fr/figures
manualxmlfrfigures_DATA = $(THE_frFIGURES)
manualhtmlfrdir = $(datadir)/genius/help/fr/html
-manualhtmlfr_DATA = fr/html/*.html
+manualhtmlfr_DATA = fr/html/ch01.html \
+ fr/html/ch02.html \
+ fr/html/ch02s02.html \
+ fr/html/ch03.html \
+ fr/html/ch03s02.html \
+ fr/html/ch03s03.html \
+ fr/html/ch04.html \
+ fr/html/ch04s02.html \
+ fr/html/ch04s03.html \
+ fr/html/ch04s04.html \
+ fr/html/ch04s05.html \
+ fr/html/ch05.html \
+ fr/html/ch05s02.html \
+ fr/html/ch05s03.html \
+ fr/html/ch05s04.html \
+ fr/html/ch05s05.html \
+ fr/html/ch05s06.html \
+ fr/html/ch05s07.html \
+ fr/html/ch06.html \
+ fr/html/ch06s02.html \
+ fr/html/ch06s03.html \
+ fr/html/ch06s04.html \
+ fr/html/ch06s05.html \
+ fr/html/ch06s06.html \
+ fr/html/ch06s07.html \
+ fr/html/ch06s08.html \
+ fr/html/ch06s09.html \
+ fr/html/ch07.html \
+ fr/html/ch07s02.html \
+ fr/html/ch07s03.html \
+ fr/html/ch07s04.html \
+ fr/html/ch07s05.html \
+ fr/html/ch07s06.html \
+ fr/html/ch08.html \
+ fr/html/ch08s02.html \
+ fr/html/ch08s03.html \
+ fr/html/ch09.html \
+ fr/html/ch10.html \
+ fr/html/ch11.html \
+ fr/html/ch11s02.html \
+ fr/html/ch11s03.html \
+ fr/html/ch11s04.html \
+ fr/html/ch11s05.html \
+ fr/html/ch11s06.html \
+ fr/html/ch11s07.html \
+ fr/html/ch11s08.html \
+ fr/html/ch11s09.html \
+ fr/html/ch11s10.html \
+ fr/html/ch11s11.html \
+ fr/html/ch11s12.html \
+ fr/html/ch11s13.html \
+ fr/html/ch11s14.html \
+ fr/html/ch11s15.html \
+ fr/html/ch11s16.html \
+ fr/html/ch11s17.html \
+ fr/html/ch11s18.html \
+ fr/html/ch11s19.html \
+ fr/html/ch11s20.html \
+ fr/html/ch12.html \
+ fr/html/ch13.html \
+ fr/html/ch13s02.html \
+ fr/html/ch13s03.html \
+ fr/html/ch13s04.html \
+ fr/html/ch14.html \
+ fr/html/index.html
manualhtmlfrfiguresdir = $(datadir)/genius/help/fr/html/figures
manualhtmlfrfigures_DATA = $(THE_frFIGURES)
+
+#########################################################
#pt_BR
+
+THE_pt_BRFIGURES = C/figures/genius_window.png C/figures/line_plot_graph.png C/figures/line_plot.png
C/figures/parametric_graph.png C/figures/parametric.png C/figures/surface_graph.png
+
manualxmlpt_BRdir = $(datadir)/genius/help/pt_BR
manualxmlpt_BR_DATA = pt_BR/genius.xml
manualxmlpt_BRfiguresdir = $(datadir)/genius/help/pt_BR/figures
-manualxmlpt_BRfigures_DATA = $(THE_CFIGURES)
+manualxmlpt_BRfigures_DATA = $(THE_pt_BRFIGURES)
manualhtmlpt_BRdir = $(datadir)/genius/help/pt_BR/html
-manualhtmlpt_BR_DATA = pt_BR/html/*.html
+manualhtmlpt_BR_DATA = pt_BR/html/ch01.html \
+ pt_BR/html/ch02.html \
+ pt_BR/html/ch02s02.html \
+ pt_BR/html/ch03.html \
+ pt_BR/html/ch03s02.html \
+ pt_BR/html/ch03s03.html \
+ pt_BR/html/ch04.html \
+ pt_BR/html/ch04s02.html \
+ pt_BR/html/ch04s03.html \
+ pt_BR/html/ch04s04.html \
+ pt_BR/html/ch04s05.html \
+ pt_BR/html/ch05.html \
+ pt_BR/html/ch05s02.html \
+ pt_BR/html/ch05s03.html \
+ pt_BR/html/ch05s04.html \
+ pt_BR/html/ch05s05.html \
+ pt_BR/html/ch05s06.html \
+ pt_BR/html/ch05s07.html \
+ pt_BR/html/ch06.html \
+ pt_BR/html/ch06s02.html \
+ pt_BR/html/ch06s03.html \
+ pt_BR/html/ch06s04.html \
+ pt_BR/html/ch06s05.html \
+ pt_BR/html/ch06s06.html \
+ pt_BR/html/ch06s07.html \
+ pt_BR/html/ch06s08.html \
+ pt_BR/html/ch06s09.html \
+ pt_BR/html/ch07.html \
+ pt_BR/html/ch07s02.html \
+ pt_BR/html/ch07s03.html \
+ pt_BR/html/ch07s04.html \
+ pt_BR/html/ch07s05.html \
+ pt_BR/html/ch07s06.html \
+ pt_BR/html/ch08.html \
+ pt_BR/html/ch08s02.html \
+ pt_BR/html/ch08s03.html \
+ pt_BR/html/ch09.html \
+ pt_BR/html/ch10.html \
+ pt_BR/html/ch11.html \
+ pt_BR/html/ch11s02.html \
+ pt_BR/html/ch11s03.html \
+ pt_BR/html/ch11s04.html \
+ pt_BR/html/ch11s05.html \
+ pt_BR/html/ch11s06.html \
+ pt_BR/html/ch11s07.html \
+ pt_BR/html/ch11s08.html \
+ pt_BR/html/ch11s09.html \
+ pt_BR/html/ch11s10.html \
+ pt_BR/html/ch11s11.html \
+ pt_BR/html/ch11s12.html \
+ pt_BR/html/ch11s13.html \
+ pt_BR/html/ch11s14.html \
+ pt_BR/html/ch11s15.html \
+ pt_BR/html/ch11s16.html \
+ pt_BR/html/ch11s17.html \
+ pt_BR/html/ch11s18.html \
+ pt_BR/html/ch11s19.html \
+ pt_BR/html/ch11s20.html \
+ pt_BR/html/ch12.html \
+ pt_BR/html/ch13.html \
+ pt_BR/html/ch13s02.html \
+ pt_BR/html/ch13s03.html \
+ pt_BR/html/ch13s04.html \
+ pt_BR/html/ch14.html \
+ pt_BR/html/index.html
manualhtmlpt_BRfiguresdir = $(datadir)/genius/help/pt_BR/html/figures
-manualhtmlpt_BRfigures_DATA = $(THE_CFIGURES)
+manualhtmlpt_BRfigures_DATA = $(THE_pt_BRFIGURES)
+
+#########################################################
#ru
+
+THE_ruFIGURES = C/figures/genius_window.png C/figures/line_plot_graph.png C/figures/line_plot.png
C/figures/parametric_graph.png C/figures/parametric.png C/figures/surface_graph.png
+
manualxmlrudir = $(datadir)/genius/help/ru
manualxmlru_DATA = ru/genius.xml
manualxmlrufiguresdir = $(datadir)/genius/help/ru/figures
-manualxmlrufigures_DATA = $(THE_CFIGURES)
+manualxmlrufigures_DATA = $(THE_ruFIGURES)
manualhtmlrudir = $(datadir)/genius/help/ru/html
-manualhtmlru_DATA = ru/html/*.html
+manualhtmlru_DATA = ru/html/ch01.html \
+ ru/html/ch02.html \
+ ru/html/ch02s02.html \
+ ru/html/ch03.html \
+ ru/html/ch03s02.html \
+ ru/html/ch03s03.html \
+ ru/html/ch04.html \
+ ru/html/ch04s02.html \
+ ru/html/ch04s03.html \
+ ru/html/ch04s04.html \
+ ru/html/ch04s05.html \
+ ru/html/ch05.html \
+ ru/html/ch05s02.html \
+ ru/html/ch05s03.html \
+ ru/html/ch05s04.html \
+ ru/html/ch05s05.html \
+ ru/html/ch05s06.html \
+ ru/html/ch05s07.html \
+ ru/html/ch06.html \
+ ru/html/ch06s02.html \
+ ru/html/ch06s03.html \
+ ru/html/ch06s04.html \
+ ru/html/ch06s05.html \
+ ru/html/ch06s06.html \
+ ru/html/ch06s07.html \
+ ru/html/ch06s08.html \
+ ru/html/ch06s09.html \
+ ru/html/ch07.html \
+ ru/html/ch07s02.html \
+ ru/html/ch07s03.html \
+ ru/html/ch07s04.html \
+ ru/html/ch07s05.html \
+ ru/html/ch07s06.html \
+ ru/html/ch08.html \
+ ru/html/ch08s02.html \
+ ru/html/ch08s03.html \
+ ru/html/ch09.html \
+ ru/html/ch10.html \
+ ru/html/ch11.html \
+ ru/html/ch11s02.html \
+ ru/html/ch11s03.html \
+ ru/html/ch11s04.html \
+ ru/html/ch11s05.html \
+ ru/html/ch11s06.html \
+ ru/html/ch11s07.html \
+ ru/html/ch11s08.html \
+ ru/html/ch11s09.html \
+ ru/html/ch11s10.html \
+ ru/html/ch11s11.html \
+ ru/html/ch11s12.html \
+ ru/html/ch11s13.html \
+ ru/html/ch11s14.html \
+ ru/html/ch11s15.html \
+ ru/html/ch11s16.html \
+ ru/html/ch11s17.html \
+ ru/html/ch11s18.html \
+ ru/html/ch11s19.html \
+ ru/html/ch11s20.html \
+ ru/html/ch12.html \
+ ru/html/ch13.html \
+ ru/html/ch13s02.html \
+ ru/html/ch13s03.html \
+ ru/html/ch13s04.html \
+ ru/html/ch14.html \
+ ru/html/index.html
manualhtmlrufiguresdir = $(datadir)/genius/help/ru/html/figures
-manualhtmlrufigures_DATA = $(THE_CFIGURES)
+manualhtmlrufigures_DATA = $(THE_ruFIGURES)
+
+#########################################################
#sv
-THE_svFIGURES = sv/figures/parametric.png \
- sv/figures/genius_window.png \
- sv/figures/line_plot_graph.png \
- sv/figures/line_plot.png \
- sv/figures/parametric_graph.png \
- sv/figures/surface_graph.png
+
+THE_svFIGURES = sv/figures/genius_window.png sv/figures/line_plot_graph.png sv/figures/line_plot.png
sv/figures/parametric_graph.png sv/figures/parametric.png sv/figures/surface_graph.png
manualxmlsvdir = $(datadir)/genius/help/sv
manualxmlsv_DATA = sv/genius.xml
@@ -142,40 +654,709 @@ manualxmlsvfiguresdir = $(datadir)/genius/help/sv/figures
manualxmlsvfigures_DATA = $(THE_svFIGURES)
manualhtmlsvdir = $(datadir)/genius/help/sv/html
-manualhtmlsv_DATA = sv/html/*.html
+manualhtmlsv_DATA = sv/html/ch01.html \
+ sv/html/ch02.html \
+ sv/html/ch02s02.html \
+ sv/html/ch03.html \
+ sv/html/ch03s02.html \
+ sv/html/ch03s03.html \
+ sv/html/ch04.html \
+ sv/html/ch04s02.html \
+ sv/html/ch04s03.html \
+ sv/html/ch04s04.html \
+ sv/html/ch04s05.html \
+ sv/html/ch05.html \
+ sv/html/ch05s02.html \
+ sv/html/ch05s03.html \
+ sv/html/ch05s04.html \
+ sv/html/ch05s05.html \
+ sv/html/ch05s06.html \
+ sv/html/ch05s07.html \
+ sv/html/ch06.html \
+ sv/html/ch06s02.html \
+ sv/html/ch06s03.html \
+ sv/html/ch06s04.html \
+ sv/html/ch06s05.html \
+ sv/html/ch06s06.html \
+ sv/html/ch06s07.html \
+ sv/html/ch06s08.html \
+ sv/html/ch06s09.html \
+ sv/html/ch07.html \
+ sv/html/ch07s02.html \
+ sv/html/ch07s03.html \
+ sv/html/ch07s04.html \
+ sv/html/ch07s05.html \
+ sv/html/ch07s06.html \
+ sv/html/ch08.html \
+ sv/html/ch08s02.html \
+ sv/html/ch08s03.html \
+ sv/html/ch09.html \
+ sv/html/ch10.html \
+ sv/html/ch11.html \
+ sv/html/ch11s02.html \
+ sv/html/ch11s03.html \
+ sv/html/ch11s04.html \
+ sv/html/ch11s05.html \
+ sv/html/ch11s06.html \
+ sv/html/ch11s07.html \
+ sv/html/ch11s08.html \
+ sv/html/ch11s09.html \
+ sv/html/ch11s10.html \
+ sv/html/ch11s11.html \
+ sv/html/ch11s12.html \
+ sv/html/ch11s13.html \
+ sv/html/ch11s14.html \
+ sv/html/ch11s15.html \
+ sv/html/ch11s16.html \
+ sv/html/ch11s17.html \
+ sv/html/ch11s18.html \
+ sv/html/ch11s19.html \
+ sv/html/ch11s20.html \
+ sv/html/ch12.html \
+ sv/html/ch13.html \
+ sv/html/ch13s02.html \
+ sv/html/ch13s03.html \
+ sv/html/ch13s04.html \
+ sv/html/ch14.html \
+ sv/html/index.html
manualhtmlsvfiguresdir = $(datadir)/genius/help/sv/html/figures
manualhtmlsvfigures_DATA = $(THE_svFIGURES)
-#
-# Text version
+
+
+#########################################################
+# Text version of the manual
manualdir = $(datadir)/genius
manual_DATA = genius.txt
+#########################################################
+# Aaaaand here's all the files ...
+
+
EXTRA_DIST = genius.txt \
- C/legal.xml \
- C/genius.xml \
- C/html/*.html \
- cs/genius.xml \
- cs/html/*.html \
- de/genius.xml \
- de/html/*.html \
- el/genius.xml \
- el/html/*.html \
- es/genius.xml \
- es/html/*.html \
- fr/genius.xml \
- fr/html/*.html \
- pt_BR/genius.xml \
- pt_BR/html/*.html \
- ru/genius.xml \
- ru/html/*.html \
- sv/genius.xml \
- sv/html/*.html \
- $(THE_CFIGURES) \
- $(THE_csFIGURES) \
- $(THE_deFIGURES) \
- $(THE_frFIGURES) \
- $(THE_svFIGURES) \
- update-po.sh \
- update-xml-to-txt-html.sh
+ C/html/ch01.html \
+ C/html/ch02.html \
+ C/html/ch02s02.html \
+ C/html/ch03.html \
+ C/html/ch03s02.html \
+ C/html/ch03s03.html \
+ C/html/ch04.html \
+ C/html/ch04s02.html \
+ C/html/ch04s03.html \
+ C/html/ch04s04.html \
+ C/html/ch04s05.html \
+ C/html/ch05.html \
+ C/html/ch05s02.html \
+ C/html/ch05s03.html \
+ C/html/ch05s04.html \
+ C/html/ch05s05.html \
+ C/html/ch05s06.html \
+ C/html/ch05s07.html \
+ C/html/ch06.html \
+ C/html/ch06s02.html \
+ C/html/ch06s03.html \
+ C/html/ch06s04.html \
+ C/html/ch06s05.html \
+ C/html/ch06s06.html \
+ C/html/ch06s07.html \
+ C/html/ch06s08.html \
+ C/html/ch06s09.html \
+ C/html/ch07.html \
+ C/html/ch07s02.html \
+ C/html/ch07s03.html \
+ C/html/ch07s04.html \
+ C/html/ch07s05.html \
+ C/html/ch07s06.html \
+ C/html/ch08.html \
+ C/html/ch08s02.html \
+ C/html/ch08s03.html \
+ C/html/ch09.html \
+ C/html/ch10.html \
+ C/html/ch11.html \
+ C/html/ch11s02.html \
+ C/html/ch11s03.html \
+ C/html/ch11s04.html \
+ C/html/ch11s05.html \
+ C/html/ch11s06.html \
+ C/html/ch11s07.html \
+ C/html/ch11s08.html \
+ C/html/ch11s09.html \
+ C/html/ch11s10.html \
+ C/html/ch11s11.html \
+ C/html/ch11s12.html \
+ C/html/ch11s13.html \
+ C/html/ch11s14.html \
+ C/html/ch11s15.html \
+ C/html/ch11s16.html \
+ C/html/ch11s17.html \
+ C/html/ch11s18.html \
+ C/html/ch11s19.html \
+ C/html/ch11s20.html \
+ C/html/ch12.html \
+ C/html/ch13.html \
+ C/html/ch13s02.html \
+ C/html/ch13s03.html \
+ C/html/ch13s04.html \
+ C/html/ch14.html \
+ C/html/index.html \
+ C/figures/genius_window.png \
+ C/figures/line_plot_graph.png \
+ C/figures/line_plot.png \
+ C/figures/parametric_graph.png \
+ C/figures/parametric.png \
+ C/figures/surface_graph.png \
+ C/genius.xml \
+ C/legal.xml \
+ cs/figures/genius_window.png \
+ cs/figures/line_plot_graph.png \
+ cs/figures/line_plot.png \
+ cs/figures/parametric_graph.png \
+ cs/figures/parametric.png \
+ cs/figures/surface_graph.png \
+ cs/html/ch01.html \
+ cs/html/ch02.html \
+ cs/html/ch02s02.html \
+ cs/html/ch03.html \
+ cs/html/ch03s02.html \
+ cs/html/ch03s03.html \
+ cs/html/ch04.html \
+ cs/html/ch04s02.html \
+ cs/html/ch04s03.html \
+ cs/html/ch04s04.html \
+ cs/html/ch04s05.html \
+ cs/html/ch05.html \
+ cs/html/ch05s02.html \
+ cs/html/ch05s03.html \
+ cs/html/ch05s04.html \
+ cs/html/ch05s05.html \
+ cs/html/ch05s06.html \
+ cs/html/ch05s07.html \
+ cs/html/ch06.html \
+ cs/html/ch06s02.html \
+ cs/html/ch06s03.html \
+ cs/html/ch06s04.html \
+ cs/html/ch06s05.html \
+ cs/html/ch06s06.html \
+ cs/html/ch06s07.html \
+ cs/html/ch06s08.html \
+ cs/html/ch06s09.html \
+ cs/html/ch07.html \
+ cs/html/ch07s02.html \
+ cs/html/ch07s03.html \
+ cs/html/ch07s04.html \
+ cs/html/ch07s05.html \
+ cs/html/ch07s06.html \
+ cs/html/ch08.html \
+ cs/html/ch08s02.html \
+ cs/html/ch08s03.html \
+ cs/html/ch09.html \
+ cs/html/ch10.html \
+ cs/html/ch11.html \
+ cs/html/ch11s02.html \
+ cs/html/ch11s03.html \
+ cs/html/ch11s04.html \
+ cs/html/ch11s05.html \
+ cs/html/ch11s06.html \
+ cs/html/ch11s07.html \
+ cs/html/ch11s08.html \
+ cs/html/ch11s09.html \
+ cs/html/ch11s10.html \
+ cs/html/ch11s11.html \
+ cs/html/ch11s12.html \
+ cs/html/ch11s13.html \
+ cs/html/ch11s14.html \
+ cs/html/ch11s15.html \
+ cs/html/ch11s16.html \
+ cs/html/ch11s17.html \
+ cs/html/ch11s18.html \
+ cs/html/ch11s19.html \
+ cs/html/ch11s20.html \
+ cs/html/ch12.html \
+ cs/html/ch13.html \
+ cs/html/ch13s02.html \
+ cs/html/ch13s03.html \
+ cs/html/ch13s04.html \
+ cs/html/ch14.html \
+ cs/html/index.html \
+ cs/genius.xml \
+ de/figures/genius_window.png \
+ de/figures/line_plot_graph.png \
+ de/figures/line_plot.png \
+ de/figures/parametric_graph.png \
+ de/figures/parametric.png \
+ de/figures/surface_graph.png \
+ de/html/ch01.html \
+ de/html/ch02.html \
+ de/html/ch02s02.html \
+ de/html/ch03.html \
+ de/html/ch03s02.html \
+ de/html/ch03s03.html \
+ de/html/ch04.html \
+ de/html/ch04s02.html \
+ de/html/ch04s03.html \
+ de/html/ch04s04.html \
+ de/html/ch04s05.html \
+ de/html/ch05.html \
+ de/html/ch05s02.html \
+ de/html/ch05s03.html \
+ de/html/ch05s04.html \
+ de/html/ch05s05.html \
+ de/html/ch05s06.html \
+ de/html/ch05s07.html \
+ de/html/ch06.html \
+ de/html/ch06s02.html \
+ de/html/ch06s03.html \
+ de/html/ch06s04.html \
+ de/html/ch06s05.html \
+ de/html/ch06s06.html \
+ de/html/ch06s07.html \
+ de/html/ch06s08.html \
+ de/html/ch06s09.html \
+ de/html/ch07.html \
+ de/html/ch07s02.html \
+ de/html/ch07s03.html \
+ de/html/ch07s04.html \
+ de/html/ch07s05.html \
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diff --git a/help/cs/html/ch05s07.html b/help/cs/html/ch05s07.html
index f261f33..8f8ec84 100644
--- a/help/cs/html/ch05s07.html
+++ b/help/cs/html/ch05s07.html
@@ -1,6 +1,53 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Seznam operátorů
GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch05.html" title="Kapitola 5.
Základy jazyka GEL"><link rel="prev" href="ch05s06.html" title="Modulární aritmetika"><link rel="next"
href="ch06.html" title="Kapitola 6. Programování s jazykem GEL"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Seznam operátorů GEL</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch05s06.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 5. Základy
jazyka GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06.html">Další</a></td></tr></table><hr></div><div class="sec
t1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-operator-list"></a>Seznam operátorů GEL</h2></div></div></div><p>Vše v jazyce GEL jsou ve
skutečnosti jen výrazy. Výrazy jsou dohromady řetězeny pomocí různých operátorů. Jak jste již viděli, i
oddělovač je ve skutečnosti jen binární operátor jazyka. Zde je seznam operátorů jazyka GEL.</p><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>Oddělovač, který vyhodnocuje jak <code
class="varname">a</code>, tak <code class="varname">b</code>, ale vrací výsledek pouze z <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>Operátor přiřazení. </p></dd><dt><span
class="term"><strong class="userinput"><code>a:=b</code></strong></span></dt><dd><p>Operátor přiřazení.
Přiřadí <code cla
ss="varname">b</code> do <code class="varname">a</code> (<code class="varname">a</code> musí být platná <a
class="link" href="ch06s09.html" title="L-hodnoty">l-hodnota</a>). Liší se od <code class="literal">=</code>,
protože se nikdy nepřevádí na <code class="literal">==</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>|a|</code></strong></span></dt><dd><p>Absolutní hodnota. V případě, že výraz je
komplexní číslo, je vrácen modul (absolutní hodnota komplexního čísla, někdy také nazýván norma), což je
vzdálenost od počátku. Například: <strong class="userinput"><code>|3 * e^(1i*pi)|</code></strong> vrátí
3.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld</a> (text je v angličtině) a
<a class="ulink" href="http://cs.wikipedia.org/wiki/Absolutn%C3%AD_hodnota";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Umocnění, umocní <code
class="varname">a</code> na <code class="varname">b</code>-tou.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Umocňování prvek po prvku. Umocní každý prvek
matice <code class="varname">a</code> na <code class="varname">b</code>-tou. Nebo, když je <code
class="varname">b</code> matice stejné velikosti jako <code class="varname">a</code>, umocňuje se prvek po
prvku. Pokud je <code class="varname">a</code> číslo a <code class="varname">b</code> je matice, pak se
vytvoří matice stejné velikosti jako <code class="varname">b</code> s <code class="varname">a</code>
umocněným na všechny různé mocnitele v <code class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a+b</code></strong></span></dt><dd><p>Sčítání. Sečte dvě čísla, matice, funkce nebo
řetězce. Pokud přičtete řetě
zec k čemukoliv, výsledkem bude vždy řetězec. Pokud je jeden operand čtvercová matice a druhý číslo, je
číslo vynásobeno jednotkovou maticí.</p></dd><dt><span class="term"><strong
class="userinput"><code>a-b</code></strong></span></dt><dd><p>Odčítání. Odečte dvě čísla, matice nebo
funkce.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Násobení. Jedná se o normální násobení
matic.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Násobení prvek po prvku v situaci, kdy <code
class="varname">a</code> a <code class="varname">b</code> jsou matice.</p></dd><dt><span class="term"><strong
class="userinput"><code>a/b</code></strong></span></dt><dd><p>Dělení. Pokud jsou <code
class="varname">a</code> a <code class="varname">b</code> čísla, jedná se o běžné dělení. Pokud to jsou
matice, odpovídá to <strong class="userinput"><code>a*b^-1</c
ode></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>Dělení prvek po prvku. Pro čísla je to stejné
jako <strong class="userinput"><code>a/b</code></strong>, ale u matic to funguje prvek po
prvku.</p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Zpětné dělení. Je to to stejné, jako <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Zpětné dělení prvků prvky.</p></dd><dt><span
class="term"><strong class="userinput"><code>a%b</code></strong></span></dt><dd><p>Operátor zbytku. Nepřepíná
do režimu <a class="link" href="ch05s06.html" title="Modulární aritmetika">modulární aritmetiky</a>, ale jen
prostě vrátí zbytek podílu <strong class="userinput"><code>a/b</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a
.%b</code></strong></span></dt><dd><p>Operátor zbytku dělení prvků prvky. Vrací zbytky po dělení
celočíselných prvků celočíselnými prvky <strong
class="userinput"><code>a./b</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a mod b</code></strong></span></dt><dd><p>Operátor modulární aritmetiky. Výraz <code
class="varname">a</code> je vyhodnocen modulární aritmetikou vůči <code class="varname">b</code>. Viz <a
class="xref" href="ch05s06.html" title="Modulární aritmetika">„Modulární aritmetika“</a>. Některé funkce a
operátory se chovají odlišně při modulární aritmetice s celými čísly.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Operátor faktoriálu. Je to jako <strong
class="userinput"><code>1*…*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Operátor dvojitého f
aktoriálu. Je to jako <strong class="userinput"><code>1*…*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a==b</code></strong></span></dt><dd><p>Operátor rovnosti, vrací
<code class="constant">true</code> (pravda) nebo <code class="constant">false</code> (nepravda) podle toho,
zda <code class="varname">a</code> je <code class="varname">b</code> rovno nebo není rovno.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Operátor nerovnosti,
vrací <code class="constant">true</code> (pravda) v případě, že <code class="varname">a</code> se nerovná
<code class="varname">b</code>, jinak vrací <code class="constant">false</code> (nepravda).</p></dd><dt><span
class="term"><strong class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Alternativní
operátor nerovnosti, vrací <code class="constant">true</code> (pravda) v případě, že <code class="varname">a</
code> se nerovná <code class="varname">b</code>, jinak vrací <code class="constant">false</code>
(nepravda).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Operátor menší než nebo rovno, vrací <code
class="constant">true</code> (pravda) v případě, že <code class="varname">a</code> je menší než nebo se rovná
<code class="varname">b</code>, jinak vrací <code class="constant">false</code> (nepravda). Je možné řetězit
ve stylu <strong class="userinput"><code>a <= b <= c</code></strong> (a může se kombinovat s operátorem
menší než).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>Operátor větší než nebo rovno, vrací <code
class="constant">true</code> (pravda) v případě, že <code class="varname">a</code> je větší než nebo se rovná
<code class="varname">b</code>, jinak vrací <code class="constant">false</code> (nepravda).
Je možné řetězit ve stylu <strong class="userinput"><code>a >= b >= c</code></strong> (a může se
kombinovat s operátorem větší než).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Operátor menší než, vrací <code
class="constant">true</code> (pravda) v případě, že <code class="varname">a</code> je menší než <code
class="varname">b</code>, jinak vrací <code class="constant">false</code> (nepravda). Je možné řetězit ve
stylu <strong class="userinput"><code>a < b < c</code></strong> (a může se kombinovat s operátorem
menší než nebo rovno).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>Operátor větší než, vrací <code
class="constant">true</code> (pravda) v případě, že <code class="varname">a</code> je větší než <code
class="varname">b</code>, jinak vrací <code class="constant">false</code> (nepravda). Je
možné řetězit ve stylu <strong class="userinput"><code>a > b > c</code></strong> (a může se
kombinovat s operátorem větší než nebo rovno).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Operátor porovnání. V případě, že <code
class="varname">a</code> je rovno <code class="varname">b</code>, vrací 0, pokud je <code
class="varname">a</code> menší než <code class="varname">b</code> vrací -1 a pokud je <code
class="varname">a</code> větší než <code class="varname">b</code>, vrací 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a and b</code></strong></span></dt><dd><p>Logické A (AND). Vrací
pravda, když <code class="varname">a</code> i <code class="varname">b</code> jsou pravda, ve všech ostatních
případech nepravda. Pokud jsou předána čísla, je se všemi nenulovými zacházeno jako s pravdivostní hodnotou
pravda.</p></dd><dt><span class="term"><strong c
lass="userinput"><code>a or b</code></strong></span></dt><dd><p>Logické NEBO (OR). Vrací pravda, když je
<code class="varname">a</code> nebo <code class="varname">b</code> (nebo oboje) pravda, jinak vrací nepravda.
Pokud jsou předána čísla, je se všemi nenulovými zacházeno jako s pravdivostní hodnotou
pravda.</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>Logické vylučovací NEBO (XOR). Vrací pravda, když právě <code
class="varname">a</code> nebo <code class="varname">b</code> pravda, ve všech ostatních případech nepravda.
Pokud jsou předána čísla, je se všemi nenulovými zacházeno jako s pravdivostní hodnotou
pravda.</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>Logická negace (NOT). Vrací logickou negaci <code
class="varname">a</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><d
d><p>Operátor negace. Vrací opačné číslo nebo matici (u matice pracuje prvek po prvku).</p></dd><dt><span
class="term"><strong class="userinput"><code>&a</code></strong></span></dt><dd><p>Reference proměnné (pro
předání odkazu na proměnnou). Viz <a class="xref" href="ch06s08.html"
title="Reference">„Reference“</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>Dereference proměnné (pro přístup k odkazované
proměnné). Viz <a class="xref" href="ch06s08.html" title="Reference">„Reference“</a>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a'</code></strong></span></dt><dd><p>Transponovat matici
komplexně sdruženou (Hermiteovsky sdružená matice). Tj. řádky a sloupce se prohodí a vezmou se komplexně
sdružená čísla ke všem prvkům. To znamená, že když prvek i,j matice <code class="varname">a</code> je x+iy,
pak prvek j,i matice <strong class="userinput"><code>a'</c
ode></strong> je x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Transponovat matici (bez komplexního sdružení).
To znamená, že prvek i,j matice <code class="varname">a</code> se stane prvkem j,i matice <strong
class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>Získat prvek matice v řádku <code
class="varname">b</code> a sloupci <code class="varname">c</code>. Pokud jsou <code class="varname">b</code>,
<code class="varname">c</code> vektory, získají se odpovídající řádky, sloupce nebo
podmatice.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Získat řádek matice (nebo více řádků, pokud
je <code class="varname">b</code> vektor).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Stejné jako
předchozí.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Získat sloupec matice (nebo sloupce, pokud
je <code class="varname">c</code> vektor).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Stejné jako předchozí.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Získat prvek z matice, s
kterou se zachází jako s vektorem. Matice se prochází řádek pro řádku.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Sestavit vektor od <code
class="varname">a</code> do <code class="varname">b</code> (nebo zadané části řádku, sloupce pro operátor
<code class="literal">@</code>). Například pro získání řádků 2 až 4 z matice <code class="varname">A</code>
byste mohli použít </p><pre class="programlisting">A@(2:4,)
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Seznam operátorů
GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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header"><tr><th colspan="3" align="center">Seznam operátorů GEL</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch05s06.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 5. Základy
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href="ch06.html">Další</a></td></tr></table><hr></div><div class="sec
t1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-operator-list"></a>Seznam operátorů GEL</h2></div></div></div><p>Vše v jazyce GEL jsou ve
skutečnosti jen výrazy. Výrazy jsou dohromady řetězeny pomocí různých operátorů. Jak jste již viděli, i
oddělovač je ve skutečnosti jen binární operátor jazyka. Zde je seznam operátorů jazyka GEL.</p><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>Oddělovač, který vyhodnocuje jak <code
class="varname">a</code>, tak <code class="varname">b</code>, ale vrací výsledek pouze z <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>Operátor přiřazení. </p></dd><dt><span
class="term"><strong class="userinput"><code>a:=b</code></strong></span></dt><dd><p>Operátor přiřazení.
Přiřadí <code cla
ss="varname">b</code> do <code class="varname">a</code> (<code class="varname">a</code> musí být platná <a
class="link" href="ch06s09.html" title="L-hodnoty">l-hodnota</a>). Liší se od <code class="literal">=</code>,
protože se nikdy nepřevádí na <code class="literal">==</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>|a|</code></strong></span></dt><dd><p>Absolutní hodnota. V případě, že výraz je
komplexní číslo, je vrácen modul (absolutní hodnota komplexního čísla, někdy také nazýván norma), což je
vzdálenost od počátku. Například: <strong class="userinput"><code>|3 * e^(1i*pi)|</code></strong> vrátí
3.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld</a> (text je v angličtině) a
<a class="ulink" href="http://cs.wikipedia.org/wiki/Absolutn%C3%AD_hodnota";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Umocnění, umocní <code
class="varname">a</code> na <code class="varname">b</code>-tou.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Umocňování prvek po prvku. Umocní každý prvek
matice <code class="varname">a</code> na <code class="varname">b</code>-tou. Nebo, když je <code
class="varname">b</code> matice stejné velikosti jako <code class="varname">a</code>, umocňuje se prvek po
prvku. Pokud je <code class="varname">a</code> číslo a <code class="varname">b</code> je matice, pak se
vytvoří matice stejné velikosti jako <code class="varname">b</code> s <code class="varname">a</code>
umocněným na všechny různé mocnitele v <code class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a+b</code></strong></span></dt><dd><p>Sčítání. Sečte dvě čísla, matice, funkce nebo
řetězce. Pokud přičtete řetě
zec k čemukoliv, výsledkem bude vždy řetězec. Pokud je jeden operand čtvercová matice a druhý číslo, je
číslo vynásobeno jednotkovou maticí.</p></dd><dt><span class="term"><strong
class="userinput"><code>a-b</code></strong></span></dt><dd><p>Odčítání. Odečte dvě čísla, matice nebo
funkce.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Násobení. Jedná se o normální násobení
matic.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Násobení prvek po prvku v situaci, kdy <code
class="varname">a</code> a <code class="varname">b</code> jsou matice.</p></dd><dt><span class="term"><strong
class="userinput"><code>a/b</code></strong></span></dt><dd><p>Dělení. Pokud jsou <code
class="varname">a</code> a <code class="varname">b</code> čísla, jedná se o běžné dělení. Pokud to jsou
matice, odpovídá to <strong class="userinput"><code>a*b^-1</c
ode></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>Dělení prvek po prvku. Pro čísla je to stejné
jako <strong class="userinput"><code>a/b</code></strong>, ale u matic to funguje prvek po
prvku.</p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Zpětné dělení. Je to to stejné, jako <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Zpětné dělení prvků prvky.</p></dd><dt><span
class="term"><strong class="userinput"><code>a%b</code></strong></span></dt><dd><p>
+ The mod operator. This does not turn on the <a class="link" href="ch05s06.html"
title="Modulární aritmetika">modular mode</a>, but
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>Operátor modulární aritmetiky. Výraz <code class="varname">a</code> je
vyhodnocen modulární aritmetikou vůči <code class="varname">b</code>. Viz <a class="xref" href="ch05s06.html"
title="Modulární aritmetika">„Modulární aritmetika“</a>. Některé funkce a operátory se chovají odlišně při
modulární aritmetice s celými čísly.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Operátor faktoriálu. Je to jako <strong
class="userinput"><code>1*…*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Operátor dvojitého faktoriálu. Je to jako
<strong class="userinput"><code>1*…*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p
Operátor rovnosti, vrací <code class="constant">true</code> (pravda) nebo <code
class="constant">false</code> (nepravda) podle toho, zda <code class="varname">a</code> je <code
class="varname">b</code> rovno nebo není rovno.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Operátor nerovnosti, vrací <code
class="constant">true</code> (pravda) v případě, že <code class="varname">a</code> se nerovná <code
class="varname">b</code>, jinak vrací <code class="constant">false</code> (nepravda).</p></dd><dt><span
class="term"><strong class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Alternativní
operátor nerovnosti, vrací <code class="constant">true</code> (pravda) v případě, že <code
class="varname">a</code> se nerovná <code class="varname">b</code>, jinak vrací <code
class="constant">false</code> (nepravda).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></str
ong></span></dt><dd><p>Operátor menší než nebo rovno, vrací <code class="constant">true</code> (pravda) v
případě, že <code class="varname">a</code> je menší než nebo se rovná <code class="varname">b</code>, jinak
vrací <code class="constant">false</code> (nepravda). Je možné řetězit ve stylu <strong
class="userinput"><code>a <= b <= c</code></strong> (a může se kombinovat s operátorem menší
než).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>
+ Greater than or equal operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than or equal to
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
+ (and they can also be combined with the greater than operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></strong></span></dt><dd><p>
+ Less than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ less than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
+ (they can also be combined with the less than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>
+ Greater than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
+ (they can also be combined with the greater than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Operátor porovnání. V případě, že <code
class="varname">a</code> je rovno <code class="varname">b</code>, vrací 0, pokud je <code
class="varname">a</code> menší než <code class="varname">b</code> vrací -1 a pokud je <code
class="varname">a</code> větší než <code class="varname">b</code>, vrací 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a and b</code></strong></span></dt><dd><p>Logické A (AND). Vrací
pravda, když <code class="varname">a</code> i <code class="varname">b</code> jsou pravda, ve všech ostatních
případech nepravda. Pokud jsou předána čísla, je se všemi nenulovými zacházeno jako s pravdivostní hodnotou
pravda.</p></dd><dt><span class="term"><strong class="userinput"><code>a or
b</code></strong></span></dt><dd><p>Logické NEBO (OR). Vrací pravda, když je <code class="varname">a</code>
nebo <
code class="varname">b</code> (nebo oboje) pravda, jinak vrací nepravda. Pokud jsou předána čísla, je se
všemi nenulovými zacházeno jako s pravdivostní hodnotou pravda.</p></dd><dt><span class="term"><strong
class="userinput"><code>a xor b</code></strong></span></dt><dd><p>
+ Logical xor.
+ Returns true if exactly one of
+ <code class="varname">a</code> or <code class="varname">b</code> is true,
+ else returns false. If given numbers, nonzero numbers
+ are treated as true.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>Operátor negace. Vrací opačné číslo nebo matici
(u matice pracuje prvek po prvku).</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Reference proměnné (pro předání odkazu na
proměnnou). Viz <a class="xref" href="ch06s08.html" title="Reference">„Reference“</a>.</p></dd><dt><span
class="term"><strong class="userinput"><code>*a</code></strong></span></dt><dd><p>Dereference proměnné (pro
přístup k odkazované proměnné). Viz <a class="xref" href="ch06s08.html"
title="Reference">„Reference“</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Transponovat matici komplexně sdruženou
(Hermiteovsky sdružená matice). Tj. řádky a sloupce se prohodí a vezmou se komplexně sdružená čísla ke všem
prvkům. To znamená, že když prvek i,
j matice <code class="varname">a</code> je x+iy, pak prvek j,i matice <strong
class="userinput"><code>a'</code></strong> je x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Transponovat matici (bez komplexního sdružení).
To znamená, že prvek i,j matice <code class="varname">a</code> se stane prvkem j,i matice <strong
class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>
+ Get element of a matrix in row <code class="varname">b</code> and column
+ <code class="varname">c</code>. If <code class="varname">b</code>,
+ <code class="varname">c</code> are vectors, then this gets the corresponding
+ rows, columns or submatrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Získat řádek matice (nebo více řádků, pokud
je <code class="varname">b</code> vektor).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Stejné jako předchozí.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Získat sloupec matice
(nebo sloupce, pokud je <code class="varname">c</code> vektor).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Stejné jako předchozí.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Získat prvek z matice, s
kterou se zachází jako s vektorem. Matice se prochází řádek pro řádku.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Sestavit vektor o
d <code class="varname">a</code> do <code class="varname">b</code> (nebo zadané části řádku, sloupce pro
operátor <code class="literal">@</code>). Například pro získání řádků 2 až 4 z matice <code
class="varname">A</code> byste mohli použít </p><pre class="programlisting">A@(2:4,)
</pre><p>, kdy <strong class="userinput"><code>2:4</code></strong> vrátí vektor <strong
class="userinput"><code>[2,3,4]</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b:c</code></strong></span></dt><dd><p>Sestavit vektor od <code
class="varname">a</code> do <code class="varname">c</code> s krokem <code class="varname">b</code>. Tj.
například </p><pre class="programlisting">genius> 1:2:9
=
`[1, 3, 5, 7, 9]
-</pre><p>Když jsou použita desetinná čísla, například <strong
class="userinput"><code>1.0:0.4:3.0</code></strong>, je výstupem to, co očekáváte, přestože se k 1,0 pětkrát
přidá 0,4, je to jen o něco více než 3,0 z důvodu, jakým jsou desetinná čísla uchována ve dvojkové soustavě
(není to přesně 0,4, ale uložené číslo je obvykle o trochu větší). Způsob, jakým je to zpracováváno, je
stejný jako u cyklu a sčítacích a násobících smyček. Pokud je konec v rámci <strong
class="userinput"><code>2^-20</code></strong>násobku velikosti kroku koncového bodu, je koncový bod použit a
předpokládá se, že nastaly chyby zaokrouhlení. To sice není perfektní, ale řeší to většinu případů. Tato
kontrola se provádí až ve verzi 1.0.18 a novějších, takže provádění vašeho kódu může být ve starších verzích
odlišné. Pokud chcete této záležitosti předejít, používejte opravdová racionální čísla, případn
ě použijte funkci <code class="function">float</code>, když si přejete na konci dostat desetinné číslo.
Například <strong class="userinput"><code>1:2/5:3</code></strong> funguje správně a <strong
class="userinput"><code>float(1:2/5:3)</code></strong> vám poskytne desetinné číslo a přitom to bude nepatrně
přesnější než <strong class="userinput"><code>1.0:0.4:3.0</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Vytvořit imaginární číslo
(vynásobit <code class="varname">a</code> imaginárním <code class="varname">i</code>). Všimněte si, že
normálně se <code class="varname">i</code> zapisuje jako <code class="varname">1i</code>. Takže předchozí je
vlastně ekvivalentní </p><pre class="programlisting">(a)*1i
- </pre></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Uvozovat identifikátor, kterýžto nebude
vyhodnocen. Nebo uvozovat matici, takže nebude rozšířena.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Přehodit hodnotu proměnné <code
class="varname">a</code> s hodnotou proměnné <code class="varname">b</code>. V současnosti nepracuje s částmi
prvků matice. Vrací <code class="constant">null</code>. Dostupné od verze 1.0.13.</p></dd><dt><span
class="term"><strong class="userinput"><code>increment a</code></strong></span></dt><dd><p>Zvýšit hodnotu
proměnné <code class="varname">a</code> o 1. V případě, že <code class="varname">a</code> je matice, je o 1
zvýšen každý prvek. Dělá to vlastně to stejné co <strong class="userinput"><code>a=a+1</code></strong>,
akorát o něco rychleji. Vrací <code class="constant">null</code>. Dostupné
od verze 1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment a by
b</code></strong></span></dt><dd><p>Zvýšit hodnotu proměnné <code class="varname">a</code> o <code
class="varname">b</code>. V případě, že <code class="varname">a</code> je matice, je o zvýšen každý prvek.
Dělá to vlastně to stejné co <strong class="userinput"><code>a=a+b</code></strong>, akorát o něco rychleji.
Vrací <code class="constant">null</code>. Dostupné od verze 1.0.13.</p></dd></dl></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Poznámka</h3><p>Operátor @() dává
operátoru : více možností. S ním můžete určovat části matice. Takže a@(2:4,6) jsou řádky 2,3,4 sloupce 6.
Nebo a@(,1:2) vám dá první dva sloupce matice. Do operátoru @() můžete i přiřazovat, stačí když je pravou
hodnotou matice o stejném rozměru jako určená oblast nebo je to jiný typ hodnoty.</p></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Poznámka</h3><p>Porovnávací operátory
(vyjma operátoru <=>, který se chová normálně) nejsou striktně binární operátory, mohou být fakticky
seskupovány běžným matematickým způsobem, např.: (1<x<=y<5) je platný pravdivostní výraz a znamená
přesně to, co by měl, tj. (1<x a x≤y a y<5)</p></div><div class="note" style="margin-left: 0.5in;
margin-right: 0.5in;"><h3 class="title">Poznámka</h3><p>Unární operátor mínus funguje různými způsoby v
závislosti na tom, kde se vyskytuje. Když se objeví před číslem, váže se přímo k němu. Když se objeví před
výrazem, má slabší vazbu než mocnina a faktoriál. Například <strong
class="userinput"><code>-1^k</code></strong> je ve skutečnosti <strong
class="userinput"><code>(-1)^k</code></strong>, ale <strong
class="userinput"><code>-neco(1)^k</code></strong> je ve skutečnosti <strong class="userinpu
t"><code>-(neco(1)^k)</code></strong>. Takže věnujte pozornost tomu, jak je používáte a pokud máte
pochybnosti, raději přidejte závorky.</p></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch05s06.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch05.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Modulární aritmetika
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
align="right" valign="top"> Kapitola 6. Programování s jazykem GEL</td></tr></table></div></body></html>
+</pre><p>Když jsou použita desetinná čísla, například <strong
class="userinput"><code>1.0:0.4:3.0</code></strong>, je výstupem to, co očekáváte, přestože se k 1,0 pětkrát
přidá 0,4, je to jen o něco více než 3,0 z důvodu, jakým jsou desetinná čísla uchována ve dvojkové soustavě
(není to přesně 0,4, ale uložené číslo je obvykle o trochu větší). Způsob, jakým je to zpracováváno, je
stejný jako u cyklu a sčítacích a násobících smyček. Pokud je konec v rámci <strong
class="userinput"><code>2^-20</code></strong>násobku velikosti kroku koncového bodu, je koncový bod použit a
předpokládá se, že nastaly chyby zaokrouhlení. To sice není perfektní, ale řeší to většinu případů. Tato
kontrola se provádí až ve verzi 1.0.18 a novějších, takže provádění vašeho kódu může být ve starších verzích
odlišné. Pokud chcete této záležitosti předejít, používejte opravdová racionální čísla, případn
ě použijte funkci <code class="function">float</code>, když si přejete na konci dostat desetinné číslo.
Například <strong class="userinput"><code>1:2/5:3</code></strong> funguje správně a <strong
class="userinput"><code>float(1:2/5:3)</code></strong> vám poskytne desetinné číslo a přitom to bude nepatrně
přesnější než <strong class="userinput"><code>1.0:0.4:3.0</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
+ written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
+ </p><pre class="programlisting">(a)*1i
+ </pre><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Uvozovat identifikátor, kterýžto nebude
vyhodnocen. Nebo uvozovat matici, takže nebude rozšířena.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Přehodit hodnotu proměnné <code
class="varname">a</code> s hodnotou proměnné <code class="varname">b</code>. V současnosti nepracuje s částmi
prvků matice. Vrací <code class="constant">null</code>. Dostupné od verze 1.0.13.</p></dd><dt><span
class="term"><strong class="userinput"><code>increment a</code></strong></span></dt><dd><p>Zvýšit hodnotu
proměnné <code class="varname">a</code> o 1. V případě, že <code class="varname">a</code> je matice, je o 1
zvýšen každý prvek. Dělá to vlastně to stejné co <strong class="userinput"><code>a=a+1</code></strong>,
akorát o něco rychleji. Vrací <code class="constant">null</code>. Dostup
né od verze 1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment a by
b</code></strong></span></dt><dd><p>Zvýšit hodnotu proměnné <code class="varname">a</code> o <code
class="varname">b</code>. V případě, že <code class="varname">a</code> je matice, je o zvýšen každý prvek.
Dělá to vlastně to stejné co <strong class="userinput"><code>a=a+b</code></strong>, akorát o něco rychleji.
Vrací <code class="constant">null</code>. Dostupné od verze 1.0.13.</p></dd></dl></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Poznámka</h3><p>Operátor @() dává
operátoru : více možností. S ním můžete určovat části matice. Takže a@(2:4,6) jsou řádky 2,3,4 sloupce 6.
Nebo a@(,1:2) vám dá první dva sloupce matice. Do operátoru @() můžete i přiřazovat, stačí když je pravou
hodnotou matice o stejném rozměru jako určená oblast nebo je to jiný typ hodnoty.</p></div><div class="no
te" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Poznámka</h3><p>Porovnávací operátory
(vyjma operátoru <=>, který se chová normálně) nejsou striktně binární operátory, mohou být fakticky
seskupovány běžným matematickým způsobem, např.: (1<x<=y<5) je platný pravdivostní výraz a znamená
přesně to, co by měl, tj. (1<x a x≤y a y<5)</p></div><div class="note" style="margin-left: 0.5in;
margin-right: 0.5in;"><h3 class="title">Poznámka</h3><p>Unární operátor mínus funguje různými způsoby v
závislosti na tom, kde se vyskytuje. Když se objeví před číslem, váže se přímo k němu. Když se objeví před
výrazem, má slabší vazbu než mocnina a faktoriál. Například <strong
class="userinput"><code>-1^k</code></strong> je ve skutečnosti <strong
class="userinput"><code>(-1)^k</code></strong>, ale <strong
class="userinput"><code>-neco(1)^k</code></strong> je ve skutečnosti <strong class="useri
nput"><code>-(neco(1)^k)</code></strong>. Takže věnujte pozornost tomu, jak je používáte a pokud máte
pochybnosti, raději přidejte závorky.</p></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch05s06.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch05.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Modulární aritmetika
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
align="right" valign="top"> Kapitola 6. Programování s jazykem GEL</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch06s05.html b/help/cs/html/ch06s05.html
index 0fbe714..50dbc67 100644
--- a/help/cs/html/ch06s05.html
+++ b/help/cs/html/ch06s05.html
@@ -1,4 +1,12 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Globální proměnné a
působnost proměnných</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch06.html" title="Kapitola 6.
Programování s jazykem GEL"><link rel="prev" href="ch06s04.html" title="Porovnávací operátory"><link
rel="next" href="ch06s06.html" title="Proměnné parametrů"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Globální proměnné a působnost proměnných</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Předcházející</a> </td><th width="60%"
align="center">Kapitola 6. Programování s jazykem GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.htm
l">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-variables-global"></a>Globální proměnné a působnost
proměnných</h2></div></div></div><p>GEL je <a class="ulink"
href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">jazyk s dynamickým rozsahem
platnosti</a>. Co to znamená hned vysvětlíme. Je to to, že normální proměnné a funkce mají dynamicky
vymezenou platnost. Výjimkou jsou <a class="link" href="ch06s06.html" title="Proměnné parametrů">proměnné
parametrů</a>, kterou jsou vždy globální.</p><p>Podobně jako většina programovacích jazyků, i GEL má různé
typy proměnných. Když je proměnná normálně definována ve funkci, je viditelná z této funkce a ze všech
funkcí, které jsou z ní volány (všechny kontexty s vyšším číslem). Například předpokládejme, že funkce <code
class="function">f</code> definuje p
roměnnou <code class="varname">a</code> a pak volá funkci <code class="function">g</code>. Potom se funkce
<code class="function">g</code> může odkazovat na proměnnou <code class="varname">a</code>. Ale jakmile dojde
k návratu z funkce <code class="function">f</code>, platnost <code class="varname">a</code> zaniká. Např.
následují kód vypíše 5. Funkce <code class="function">g</code> nemůže být volána z nejvyšší úrovně (mimo
funkci <code class="function">f</code>, protože proměnná <code class="varname">a</code> pak není definována).
</p><pre class="programlisting">function f() = (a:=5; g());
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Globální proměnné a
působnost proměnných</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch06.html" title="Kapitola 6.
Programování s jazykem GEL"><link rel="prev" href="ch06s04.html" title="Porovnávací operátory"><link
rel="next" href="ch06s06.html" title="Proměnné parametrů"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Globální proměnné a působnost proměnných</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Předcházející</a> </td><th width="60%"
align="center">Kapitola 6. Programování s jazykem GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.htm
l">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-variables-global"></a>Globální proměnné a působnost
proměnných</h2></div></div></div><p>
+ GEL is a
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ dynamically scoped language</a>. We will explain what this
+ means below. That is, normal variables and functions are dynamically
+ scoped. The exception are
+ <a class="link" href="ch06s06.html" title="Proměnné parametrů">parameter variables</a>,
+ which are always global.
+ </p><p>Podobně jako většina programovacích jazyků, i GEL má různé typy proměnných. Když je proměnná
normálně definována ve funkci, je viditelná z této funkce a ze všech funkcí, které jsou z ní volány (všechny
kontexty s vyšším číslem). Například předpokládejme, že funkce <code class="function">f</code> definuje
proměnnou <code class="varname">a</code> a pak volá funkci <code class="function">g</code>. Potom se funkce
<code class="function">g</code> může odkazovat na proměnnou <code class="varname">a</code>. Ale jakmile dojde
k návratu z funkce <code class="function">f</code>, platnost <code class="varname">a</code> zaniká. Např.
následují kód vypíše 5. Funkce <code class="function">g</code> nemůže být volána z nejvyšší úrovně (mimo
funkci <code class="function">f</code>, protože proměnná <code class="varname">a</code> pak není definována).
</p><pre class="programlisting">function f() = (a:=5; g());
function g() = print(a);
f();
</pre><p>Pokud definujete proměnnou uvnitř funkce, přepíše jinou proměnnou definovanou ve volající funkci.
Například upravíme předchozí kód a napíšeme: </p><pre class="programlisting">function f() = (a:=5; g());
diff --git a/help/cs/html/ch07s02.html b/help/cs/html/ch07s02.html
index 00ebf0d..711839c 100644
--- a/help/cs/html/ch07s02.html
+++ b/help/cs/html/ch07s02.html
@@ -1,4 +1,35 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Syntaxe v nejvyšší
úrovni</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch07.html" title="Kapitola 7.
Pokročilé programování v jazyce GEL"><link rel="prev" href="ch07.html" title="Kapitola 7. Pokročilé
programování v jazyce GEL"><link rel="next" href="ch07s03.html" title="Vracení funkcí"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Syntaxe v nejvyšší
úrovni</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch07.html">Předcházející</a> </td><th
width="60%" align="center">Kapitola 7. Pokročilé programování v jazyce GEL</th><td width="20%" align="right">
<a accesskey="n" href="ch07s03.h
tml">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-toplevel-syntax"></a>Syntaxe v nejvyšší
úrovni</h2></div></div></div><p>Syntaxe se lehce liší, když zadáváte příkazy v nejvyšší úrovni a když jsou
uvnitř závorek nebo uvnitř funkce. Na nejvyšší úrovni zadání funguje stejně, jako když zmáčknete Enter na
příkazovém řádku. Proto uvažujte o programu, jako o sekvenci řádků, které byste zadávali na příkazovém řádku.
Především nepotřebujete zadávat oddělovač na konci řádku (ledaže se jedná o část několika příkazů v
závorkách).</p><p>Následující kód skončí chybou, pokud jej zadáte v nejvyšší úrovni programu, zatímco ve
funkci bude pracovat bez problémů. </p><pre class="programlisting">if Neco() then
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Syntaxe v nejvyšší
úrovni</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch07.html" title="Kapitola 7.
Pokročilé programování v jazyce GEL"><link rel="prev" href="ch07.html" title="Kapitola 7. Pokročilé
programování v jazyce GEL"><link rel="next" href="ch07s03.html" title="Vracení funkcí"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Syntaxe v nejvyšší
úrovni</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch07.html">Předcházející</a> </td><th
width="60%" align="center">Kapitola 7. Pokročilé programování v jazyce GEL</th><td width="20%" align="right">
<a accesskey="n" href="ch07s03.h
tml">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-toplevel-syntax"></a>Syntaxe v nejvyšší
úrovni</h2></div></div></div><p>
+ The syntax is slightly different if you enter statements on
+ the top level versus when they are inside parentheses or
+ inside functions. On the top level, enter acts the same as if
+ you press return on the command line. Therefore think of programs
+ as just a sequence of lines as if they were entered on the command line.
+ In particular, you do not need to enter the separator at the end of the
+ line (unless it is of course part of several statements inside
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
+ </p><p>Následující kód skončí chybou, pokud jej zadáte v nejvyšší úrovni programu, zatímco ve funkci
bude pracovat bez problémů. </p><pre class="programlisting">if Neco() then
UdelatNeco()
else
UdelatNecoJineho()
diff --git a/help/cs/html/ch11s04.html b/help/cs/html/ch11s04.html
index c4689a2..9c76f0b 100644
--- a/help/cs/html/ch11s04.html
+++ b/help/cs/html/ch11s04.html
@@ -1 +1,28 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Konstanty</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s03.html" title="Parametry"><link rel="next" href="ch11s05.html" title="Práce s čísly"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Konstanty</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s03.html">Předcházející</a> </td><th width="60%"
align="center">Kapitola 11. Seznam funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-gel-function-list-constants"></a>Konstanty</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Catalanova konstanta, přibližně 0,915… Je definována jako řada se
členy <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, kde <code class="varname">k</code>
je z intervalu 0 až nekonečno.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/CatalansConstant.html"; target="_top">Mathworld</a> (text je v angličtině)
a <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant"; target="_top">Wikipedia</a> (text
je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alternativní názvy: <code
class="function">gamma</code></p><p>Eulerova konstanta gama. Někdy nazývaná také Eulerova-Mascheroniho
konstanta.</p><p>Více informací najdete v encyklopediíc <a class="ulink"
href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html"; target="_top">Mathworld</a>
(text je v angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Eulerova_konstanta";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre
class="synopsis">GoldenRatio</pre><p>Zlatý řez.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> (text je v angličtině),
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="ht
tp://cs.wikipedia.org/wiki/Zlat%C3%BD_%C5%99ez" target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Tíhové zrychlení na úrovni moře v metrech za sekundu na druhou. Jedná se o
standardní gravitační konstantu 9,80665. Gravitace v končinách vašeho lesa se může lišit, kvůli jiné
nadmořské výšce a kvůli tomu, že Země není ideálně kulatá a jednolitá.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="http://cs.wikipedia.org/wiki/Gravita%C4%8Dn%C3%AD_konstanta";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-e"></a>e</span></dt><dd><pre class="synopsis">e</pre><p>Základ přirozeného logaritmu.
<strong class="userinput"><code>e^x</code></strong> je exponenciální funkce <a class="link"
href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a>. Hodnota konstanty je přibližně
2,71
828182846… Toto číslo bývá někdy nazýváno Eulerovo, ačkoliv existuje několik čísel rovněž nazývaných
Eulerovo. Například konstanta gamma: <a class="link" href="ch11s04.html#gel-function-EulerConstant"><code
class="function">EulerConstant</code></a>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/E"; target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Eulerovo_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-pi"></a>pi</span></dt><dd><pre class="synopsis">pi</pre><p>Číslo pí, což je poměr obvodu
kružnice vůči jejímu průměru. Přibližně to je 3,14159265359…</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a>
(text je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/Pi.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/P%C3%AD_%28%C4%8D%C3%ADslo%29";
target="_top">Wikipedia</a>.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s03.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Parametry </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%" align="right"
valign="top"> Práce s čísly</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Konstanty</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s03.html" title="Parametry"><link rel="next" href="ch11s05.html" title="Práce s čísly"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Konstanty</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s03.html">Předcházející</a> </td><th width="60%"
align="center">Kapitola 11. Seznam funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-gel-function-list-constants"></a>Konstanty</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Catalanova konstanta, přibližně 0,915… Je definována jako řada se
členy <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, kde <code class="varname">k</code>
je z intervalu 0 až nekonečno.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alternativní názvy: <code class="function">gamma</code></p><p>Eulerova
konstanta gama. Někdy nazývaná také Eulerova-Mascheroniho konstanta.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre
class="synopsis">GoldenRatio</pre><p>Zlatý řez.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Tíhové zrychlení na úrovni moře v metrech za sekundu na druhou. Jedná se o
standardní gravitační konstantu 9,80665. Gravitace v končinách vašeho lesa se může lišit, kvůli jiné
nadmořské výšce a kvůli tomu, že Země není ideálně kulatá a jednolitá.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>Základ přirozeného logaritmu. <strong class="userinput"><code>e^x</code></strong>
je exponenciální funkce <a class="link" href="ch11s05.html#gel-function-exp"><code
class="function">exp</code></a>. Hodnota konstanty je přibližně 2,71828182846… Toto číslo bývá někdy nazýváno
Eulerovo, ačkoliv existuje několik čísel rovněž nazývaných Eulerovo. Například konstanta gamma: <a
class="link" href="ch11s04.html#gel-function-EulerConstant"><code
class="function">EulerConstant</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>Číslo pí, což je poměr obvodu kružnice vůči jejímu průměru. Přibližně to je
3,14159265359…</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s05.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Parametry
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
align="right" valign="top"> Práce s čísly</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s05.html b/help/cs/html/ch11s05.html
index 183285b..943fb85 100644
--- a/help/cs/html/ch11s05.html
+++ b/help/cs/html/ch11s05.html
@@ -1,8 +1,44 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Práce s
čísly</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11.
Seznam funkcí GEL"><link rel="prev" href="ch11s04.html" title="Konstanty"><link rel="next"
href="ch11s06.html" title="Trigonometrie"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Práce s čísly</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s04.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class
="title" style="clear: both"><a name="genius-gel-function-list-numeric"></a>Práce s
čísly</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Alternativní názvy: <code class="function">abs</code></p><p>Absolutní hodnota čísla <code
class="varname">x</code>, případně modul v případě komplexního čísla <code class="varname">x</code>. U
komplexního čísla to je vlastně vzdálenost <code class="varname">x</code> od počátku. Je to to stejné, jako
<strong class="userinput"><code>|x|</code></strong>.</p><p>Více informací najdete v encyklopedii <a
class="ulink" href="http://cs.wikipedia.org/wiki/Absolutn%C3%AD_hodnota"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolutní hodnota; text je
v angličtině)</a>, <a class="ulink" hre
f="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath (absolutní hodnota komplexního
čísla; text je v angličtině)</a>, <a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html";
target="_top">Mathworld (absolutní hodnota; text je v angličtině)</a> a <a class="ulink"
href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld (absolutní hodnota
komplexního čísla; text je v angličtině)</a>.</p></dd><dt><span class="term"><a
name="gel-function-Chop"></a>Chop</span></dt><dd><pre class="synopsis">Chop (x)</pre><p>Nahrazovat velmi malá
čísla nulou.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alternativní názvy: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Vypočítá komplexně sdružené číslo ke komplexnímu číslu <code
class="varname">z</code>. Pokud je <code
class="varname">z</code> vektor nebo matice, vypočítají se komplexně sdružená čísla pro všechny
prvky.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Komplexn%C4%9B_sdru%C5%BEen%C3%A9_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Získat jmenovatel racionálního čísla.</p><p>Více informací najdete v encyklopedii <a
class="ulink" href="http://cs.wikipedia.org/wiki/Jmenovatel"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre
class="synopsis">FractionalPart (x)</pre><p>Vrátit část čísla za desetinnou čárkou.</p><p>Více informací
najdete v encyklopedii <a class="ulink" href="http://en.wikipedia.org/wiki/Fractional_part";
target="_top">Wikipedia</a> (text je v angličtině).</p></d
d><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre class="synopsis">Im
(z)</pre><p>Alternativní názvy: <code class="function">ImaginaryPart</code></p><p>Vrátit imaginární část
komplexního čísla. Například <strong class="userinput"><code>Re(3+4i)</code></strong> vyplodí 4.</p><p>Více
informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Imagin%C3%A1rn%C3%AD_%C4%8D%C3%A1st#Z.C3.A1pis_a_souvisej.C3.ADc.C3.AD_pojmy";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Dělit beze zbytku.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Zkontrolovat, jestli je argument komplexní (ne reálné) číslo. Tím se míní opravdu číslo, které
není reálné. Takže <strong class="userinput">
<code>IsComplex(3)</code></strong> vrátí <code class="constant">false</code>, zatímco <strong
class="userinput"><code>IsComplex(3-1i)</code></strong> vrátí <code
class="constant">true</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Zkontrolovat, zda je argument komplexní racionální číslo.
Tzn., že jak reální, tak imaginární část jsou zadány jako racionální čísla. Racionálním se samozřejmě myslí,
že „není uloženo jako desetinné číslo s plovoucí čárkou“.</p></dd><dt><span class="term"><a
name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre class="synopsis">IsFloat
(num)</pre><p>Zkontrolovat, zda je argument reálné desetinné číslo (ne komplexní).</p></dd><dt><span
class="term"><a name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre
class="synopsis">IsGaussInteger (num)</pre><p>Alterna
tivní názvy: <code class="function">IsComplexInteger</code></p><p>Zkontrolovat, jestli je argument celé
komplexní číslo. Celé komplexní číslo je číslo ve tvaru <strong
class="userinput"><code>n+1i*m</code></strong>, kde <code class="varname">n</code> a <code
class="varname">m</code> jsou celá čísla.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Zkontrolovat, zda je argument celé číslo (ne komplexní).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Zkontrolovat, zda je argument nezáporné reálné celé
číslo. Tj. buď kladné celé číslo nebo nula.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Alternativní názvy: <code cl
ass="function">IsNaturalNumber</code></p><p>Zkontrolovat, zda je argument kladné reálné celé číslo.
Upozorňujeme, že se řídíme konvencí, že 0 není přirozené číslo.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Zkontrolovat, zda je argument racionální (ne komplexní) číslo. Racionální samozřejmě prostě
znamená „není uloženo jako desetinné číslo s plovoucí čárkou“.</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (num)</pre><p>Zkontrolovat,
zda je argument reálné číslo.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Získat
čitatel racionálního čísla.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/%C4%8Citatel"; target="_top">Wikip
edia</a>.</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Alternativní názvy: <code class="function">RealPart</code></p><p>Vrátit
reálnou část komplexního čísla. Například <strong class="userinput"><code>Re(3+4i)</code></strong> vyplodí
3.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Re%C3%A1ln%C3%A1_%C4%8D%C3%A1st#Z.C3.A1pis_a_souvisej.C3.ADc.C3.AD_pojmy";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Sign"></a>Sign</span></dt><dd><pre class="synopsis">Sign (x)</pre><p>Alternativní názvy:
<code class="function">sign</code></p><p>Vrátit znaménko čísla. Konkrétně vrací <code
class="literal">-1</code> u záporných čísel, <code class="literal">0</code> pro nulu a <code
class="literal">1</code> u kladných čísel. Pokud je <code class="varname">x</code> komplexní hodnota, pak
<code class="function">
Sign</code> vrací směr nebo 0.</p></dd><dt><span class="term"><a
name="gel-function-ceil"></a>ceil</span></dt><dd><pre class="synopsis">ceil (x)</pre><p>Alternativní názvy:
<code class="function">Ceiling</code></p><p>Získat nejnižší celé číslo, které je větší nebo rovno <code
class="varname">n</code>. Například: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ceil(1.1)</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Práce s
čísly</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11.
Seznam funkcí GEL"><link rel="prev" href="ch11s04.html" title="Konstanty"><link rel="next"
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header"><tr><th colspan="3" align="center">Práce s čísly</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s04.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class
="title" style="clear: both"><a name="genius-gel-function-list-numeric"></a>Práce s
čísly</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Alternativní názvy: <code class="function">abs</code></p><p>Absolutní hodnota čísla <code
class="varname">x</code>, případně modul v případě komplexního čísla <code class="varname">x</code>. U
komplexního čísla to je vlastně vzdálenost <code class="varname">x</code> od počátku. Je to to stejné, jako
<strong class="userinput"><code>|x|</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
+ <a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld
(complex modulus)</a>
+for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Nahrazovat velmi malá čísla nulou.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alternativní názvy: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Vypočítá komplexně sdružené číslo ke komplexnímu číslu <code
class="varname">z</code>. Pokud je <code class="varname">z</code> vektor nebo matice, vypočítají se komplexně
sdružená čísla pro všechny prvky.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Získat jmenovatel racionálního čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Vrátit část čísla za desetinnou čárkou.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alternativní názvy: <code class="function">ImaginaryPart</code></p><p>Vrátit
imaginární část komplexního čísla. Například <strong class="userinput"><code>Re(3+4i)</code></strong> vyplodí
4.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Dělit beze zbytku.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Zkontrolovat, jestli je argument komplexní (ne reálné) číslo. Tím se míní opravdu číslo, které
není reálné. Takže <strong class="userinput"><code>IsComplex(3)</code></strong> vrátí <code
class="constant">false</code>, zatímco <strong class="userinput"><code>IsComplex(3-1i)</code></strong> vrátí
<code class="constant">true</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Zkontrolovat, zda je argument komplexní racionální číslo.
Tzn., že jak reální, tak imaginární část jsou zadány jako racionální čís
la. Racionálním se samozřejmě myslí, že „není uloženo jako desetinné číslo s plovoucí
čárkou“.</p></dd><dt><span class="term"><a name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre
class="synopsis">IsFloat (num)</pre><p>Zkontrolovat, zda je argument reálné desetinné číslo (ne
komplexní).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(num)</pre><p>Alternativní názvy: <code class="function">IsComplexInteger</code></p><p>Zkontrolovat, jestli
je argument celé komplexní číslo. Celé komplexní číslo je číslo ve tvaru <strong
class="userinput"><code>n+1i*m</code></strong>, kde <code class="varname">n</code> a <code
class="varname">m</code> jsou celá čísla.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Zkontrolovat, zda je argument celé číslo (ne komp
lexní).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Zkontrolovat, zda je argument nezáporné reálné celé
číslo. Tj. buď kladné celé číslo nebo nula.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Alternativní názvy: <code
class="function">IsNaturalNumber</code></p><p>Zkontrolovat, zda je argument kladné reálné celé číslo.
Upozorňujeme, že se řídíme konvencí, že 0 není přirozené číslo.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Zkontrolovat, zda je argument racionální (ne komplexní) číslo. Racionální samozřejmě prostě
znamená „není uloženo jako desetinné číslo s plovoucí čárkou“.</p></
dd><dt><span class="term"><a name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre
class="synopsis">IsReal (num)</pre><p>Zkontrolovat, zda je argument reálné číslo.</p></dd><dt><span
class="term"><a name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator
(x)</pre><p>Získat čitatel racionálního čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Alternativní názvy: <code class="function">RealPart</code></p><p>Vrátit
reálnou část komplexního čísla. Například <strong class="userinput"><code>Re(3+4i)</code></strong> vyplodí
3.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Alternativní názvy: <code class="function">sign</code></p><p>Vrátit
znaménko čísla. Konkrétně vrací <code class="literal">-1</code> u záporných čísel, <code
class="literal">0</code> pro nulu a <code class="literal">1</code> u kladných čísel. Pokud je <code
class="varname">x</code> komplexní hodnota, pak <code class="function">Sign</code> vrací směr nebo
0.</p></dd><dt><span class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre
class="synopsis">ceil (x)</pre><p>Alternativní názvy: <code class="function">Ceiling</code></p><p>Získat
nejnižší celé číslo, které je větší nebo rovno <code class="varname">n</code>. Například: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ceil(1.1)</code></strong>
= 2
<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1.1)</code></strong>
= -1
-</pre><p>Měli byste být obezřetní a uvědomit si, že desetinná čísla jsou uchovávána v binární podobě, takže
nemusí mít naprosto přesně tu hodnotu, kterou očekáváte. Například <strong
class="userinput"><code>ceil(420/4.2)</code></strong> vrací 101 a ne 100, jak byste asi očekávali. To je tím,
že 4,2 je ve skutečnosti uloženo jako nepatrně méně než 4,2. Pokud chcete přesné výsledky, použijte
racionální vyjádření <strong class="userinput"><code>42/10</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-exp"></a>exp</span></dt><dd><pre class="synopsis">exp (x)</pre><p>Exponenciální funkce.
Jedná se o funkci <strong class="userinput"><code>e^x</code></strong>, kde <code class="varname">e</code> je
<a class="link" href="ch11s04.html#gel-function-e">základ přirozeného logaritmu</a>.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://planetmath.org/LogarithmFunction";
target="_top">Planetma
th</a> (text je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Exponenci%C3%A1ln%C3%AD_funkce#Exponenci.C3.A1la_o_z.C3.A1kladu_e";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-float"></a>float</span></dt><dd><pre class="synopsis">float (x)</pre><p>Udělá z čísla
desetinné číslo. Tzn., že vrací hodnotu čísla <code class="varname">x</code> v podobě čísla s plovoucí
desetinnou čárkou.</p></dd><dt><span class="term"><a name="gel-function-floor"></a>floor</span></dt><dd><pre
class="synopsis">floor (x)</pre><p>Alternativní názvy: <code class="function">Floor</code></p><p>Vrátit
nejvyšší celé číslo, které je menší nebo rovno <code class="varname">n</code>.</p></dd><dt><span
class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre class="synopsis">ln (x)</pre
<p>Přirozený logaritmus, logaritmus o základu <code class="varname">e</code>.</p><p>Více informací najdete
v encyklopediích <a class="ulink" href="http://planetmath.org/LogarithmFunction";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/NaturalLogarithm.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Logaritmus#P.C5.99irozen.C3.BD_logaritmus";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-log"></a>log</span></dt><dd><pre class="synopsis">log (x)</pre><pre class="synopsis">log
(x,b)</pre><p>Logaritmus <code class="varname">x</code> o základu <code class="varname">b</code> (v režimu
modulární aritmetiky nazýván <a class="link" href="ch11s07.html#gel-function-DiscreteLog"><code
class="function">DiscreteLog</code></a>), pokud není základ uveden, použije se <a class="link"
href="ch11s04.html#gel-function-e"><
code class="varname">e</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logaritmus čísla
<code class="varname">x</code> o základu 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Alternativní názvy:
<code class="function">lg</code></p><p>Logaritmus čísla <code class="varname">x</code> o základu
2.</p></dd><dt><span class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synopsis">max
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Vrací maximum z argumentů nebo matice.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Vrátit mini
mum z argumentů nebo matice.</p></dd><dt><span class="term"><a
name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand (velikost...)</pre><p>Generovat
náhodné desetinné číslo z intervalu <code class="literal">[0,1)</code>. Pokud je zadána velikost, pak se
vygeneruje matice (zadána dvě čísla) nebo vektor (zadáno jedno číslo) této velikosti.</p></dd><dt><span
class="term"><a name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,velikost...)</pre><p>Generovat náhodné číslo z intervalu <code class="literal">[0,max)</code>. Pokud je
zadána velikost, pak se vygeneruje matice (zadána dvě čísla) nebo vektor (zadáno jedno číslo) této velikosti.
Například </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
+</pre><p>Měli byste být obezřetní a uvědomit si, že desetinná čísla jsou uchovávána v binární podobě, takže
nemusí mít naprosto přesně tu hodnotu, kterou očekáváte. Například <strong
class="userinput"><code>ceil(420/4.2)</code></strong> vrací 101 a ne 100, jak byste asi očekávali. To je tím,
že 4,2 je ve skutečnosti uloženo jako nepatrně méně než 4,2. Pokud chcete přesné výsledky, použijte
racionální vyjádření <strong class="userinput"><code>42/10</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-exp"></a>exp</span></dt><dd><pre class="synopsis">exp (x)</pre><p>Exponenciální funkce.
Jedná se o funkci <strong class="userinput"><code>e^x</code></strong>, kde <code class="varname">e</code> je
<a class="link" href="ch11s04.html#gel-function-e">základ přirozeného logaritmu</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Udělá z čísla desetinné číslo. Tzn., že vrací hodnotu čísla <code
class="varname">x</code> v podobě čísla s plovoucí desetinnou čárkou.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Alternativní
názvy: <code class="function">Floor</code></p><p>Vrátit nejvyšší celé číslo, které je menší nebo rovno <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>Přirozený logaritmus, logaritmus o základu <code
class="varname">e</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logaritmus <code
class="varname">x</code> o základu <code class="varname">b</code> (v režimu modulární aritmetiky nazýván <a
class="link" href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a>),
pokud není základ uveden, použije se <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logaritmus čísla
<code class="varname">x</code> o základu 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Alternativní názvy:
<code class="function">lg</code></p><p>Logaritmus čísla <code class="varname">x</code> o základu
2.</p></dd><dt><span class="ter
m"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synopsis">max
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Vrací maximum z argumentů nebo matice.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Vrátit minimum z argumentů nebo matice.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(velikost...)</pre><p>Generovat náhodné desetinné číslo z intervalu <code class="literal">[0,1)</code>. Pokud
je zadána velikost, pak se vygeneruje matice (zadána dvě čísla) nebo vektor (zadáno jedno číslo) této
velikosti.</p></dd><dt><span class="term"><a name="gel-function-randint"></a>randint</span></dt><dd><pre
class="synopsis">randint (max
,velikost...)</pre><p>Generovat náhodné číslo z intervalu <code class="literal">[0,max)</code>. Pokud je
zadána velikost, pak se vygeneruje matice (zadána dvě čísla) nebo vektor (zadáno jedno číslo) této velikosti.
Například </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
= 3
<code class="prompt">genius></code> <strong class="userinput"><code>randint(4,2)</code></strong>
=
diff --git a/help/cs/html/ch11s06.html b/help/cs/html/ch11s06.html
index 49719b3..baf23b5 100644
--- a/help/cs/html/ch11s06.html
+++ b/help/cs/html/ch11s06.html
@@ -1,2 +1,34 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometrie</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Příručka k aplikaci Genius"><link rel="up"
href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev" href="ch11s05.html" title="Práce s
čísly"><link rel="next" href="ch11s07.html" title="Teorie čísel"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometrie</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 cla
ss="title" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonometrie</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alternativní názvy:
<code class="function">arccos</code></p><p>Funkce arkus kosinus (inverzní kosinus).</p></dd><dt><span
class="term"><a name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh
(x)</pre><p>Alternativní názvy: <code class="function">arccosh</code></p><p>Funkce arkus hyperbolický kosinus
(inverzní cosh).</p></dd><dt><span class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre
class="synopsis">acot (x)</pre><p>Alternativní názvy: <code class="function">arccot</code></p><p>Funkce arkus
kotangens (inverzní kotangens).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre
<p>Alternativní názvy: <code class="function">arccoth</code></p><p>Funkce arkus hyperbolický kotangens
(inverzní coth).</p></dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre
class="synopsis">acsc (x)</pre><p>Alternativní názvy: <code class="function">arccsc</code></p><p>Funkce
inverzní kosekans.</p></dd><dt><span class="term"><a
name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch (x)</pre><p>Alternativní
názvy: <code class="function">arccsch</code></p><p>Funkce inverzní hyperbolický kosekans.</p></dd><dt><span
class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec
(x)</pre><p>Alternativní názvy: <code class="function">arcsec</code></p><p>Funkce inverzní
sekans.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Alternativní názvy: <code class="function">arcsech</code></p><p>Funkce
inverzn�
� hyperbolický sekans.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin (x)</pre><p>Alternativní názvy:
<code class="function">arcsin</code></p><p>Funkce arkus sinus (inverzní sinus).</p></dd><dt><span
class="term"><a name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh
(x)</pre><p>Alternativní názvy: <code class="function">arcsinh</code></p><p>Funkce arkus hyperbolický sinus
(inverzní sinh).</p></dd><dt><span class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre
class="synopsis">atan (x)</pre><p>Alternativní názvy: <code class="function">arctan</code></p><p>Vypočítat
funkce arkus tangens (inverzní tangens).</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> (text je v angličtině) a
<a class="ulink" href="http://cs.wikipedia.org/wiki/Arkus_tangens"; target="_top"
Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alternativní názvy: <code class="function">arctanh</code></p><p>Funkce
arkus hyperbolický tangens (inverzní tanh).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alternativní
názvy: <code class="function">arctan2</code></p><p>Vypočítat funkci arctan2. Jestliže je <strong
class="userinput"><code>x>0</code></strong>, pak vrací <strong
class="userinput"><code>atan(y/x)</code></strong>. Jestliže je <strong
class="userinput"><code>x<0</code></strong>, vrací <strong class="userinput"><code>sign(y) * (pi -
atan(|y/x|)</code></strong>. A při <strong class="userinput"><code>x=0</code></strong> vrací <strong
class="userinput"><code>sign(y) *
- pi/2</code></strong>. Volání <strong class="userinput"><code>atan2(0,0)</code></strong> vrací 0
namísto selhání.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> (text je v angličtině) a
<a class="ulink" href="http://cs.wikipedia.org/wiki/Arctg2"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre class="synopsis">cos
(x)</pre><p>Vypočítat funkci kosinus.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce"; target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Vypočítat funkci hyperbolický kosinus.</p><
p>Více informací najdete v encyklopediích <a class="ulink"
href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>Funkce kotangens.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce"; target="_top">Wikipedia</a> a <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Funkce hyperbolický kotangens.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_
top">Wikipedia</a> a <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-csc"></a>csc</span></dt><dd><pre class="synopsis">csc (x)</pre><p>Funkce
kosekans.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce"; target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Funkce hyperbolický kosekans.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličt
ině).</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Funkce sekans.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce"; target="_top">Wikipedia</a> a <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Funkce hyperbolický sekans.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Vypočítat f
unkci sinus.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://cs.wikipedia.org/wiki/Goniometrick%C3%A1_funkce"; target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Vypočítat funkci hyperbolický sinus.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce";
target="_top">Wikipedia</a> a <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-tan"></a>tan</span></dt><dd><pre class="synopsis">tan (x)</pre><p>Vypočítat funkci
tangens.</p><p>Více informací najdete v encyklopediích <a class="ulink" href="https://cs.wikipedia.org/wiki/
Goniometrick%C3%A1_funkce" target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Funkce hyperbolický tangens.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s05.html">Předcházející</a> </td><td
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+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometrie</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Příručka k aplikaci Genius"><link rel="up"
href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev" href="ch11s05.html" title="Práce s
čísly"><link rel="next" href="ch11s07.html" title="Teorie čísel"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometrie</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 cla
ss="title" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonometrie</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alternativní názvy:
<code class="function">arccos</code></p><p>Funkce arkus kosinus (inverzní kosinus).</p></dd><dt><span
class="term"><a name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh
(x)</pre><p>Alternativní názvy: <code class="function">arccosh</code></p><p>Funkce arkus hyperbolický kosinus
(inverzní cosh).</p></dd><dt><span class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre
class="synopsis">acot (x)</pre><p>Alternativní názvy: <code class="function">arccot</code></p><p>Funkce arkus
kotangens (inverzní kotangens).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre
<p>Alternativní názvy: <code class="function">arccoth</code></p><p>Funkce arkus hyperbolický kotangens
(inverzní coth).</p></dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre
class="synopsis">acsc (x)</pre><p>Alternativní názvy: <code class="function">arccsc</code></p><p>Funkce
inverzní kosekans.</p></dd><dt><span class="term"><a
name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch (x)</pre><p>Alternativní
názvy: <code class="function">arccsch</code></p><p>Funkce inverzní hyperbolický kosekans.</p></dd><dt><span
class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec
(x)</pre><p>Alternativní názvy: <code class="function">arcsec</code></p><p>Funkce inverzní
sekans.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Alternativní názvy: <code class="function">arcsech</code></p><p>Funkce
inverzn�
� hyperbolický sekans.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin (x)</pre><p>Alternativní názvy:
<code class="function">arcsin</code></p><p>Funkce arkus sinus (inverzní sinus).</p></dd><dt><span
class="term"><a name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh
(x)</pre><p>Alternativní názvy: <code class="function">arcsinh</code></p><p>Funkce arkus hyperbolický sinus
(inverzní sinh).</p></dd><dt><span class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre
class="synopsis">atan (x)</pre><p>Alternativní názvy: <code class="function">arctan</code></p><p>Vypočítat
funkce arkus tangens (inverzní tangens).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alternativní názvy: <code class="function">arctanh</code></p><p>Funkce
arkus hyperbolický tangens (inverzní tanh).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alternativní
názvy: <code class="function">arctan2</code></p><p>Vypočítat funkci arctan2. Jestliže je <strong
class="userinput"><code>x>0</code></strong>, pak vrací <strong
class="userinput"><code>atan(y/x)</code></strong>. Jestliže je <strong
class="userinput"><code>x<0</code></strong>, vrací <strong class="userinput"><code>sign(y) * (pi -
atan(|y/x|)</code></strong>. A při <strong class="userinput"><code>x=0</code></strong> vrací <strong
class="userinput"><code>sign(y) *
+ pi/2</code></strong>. Volání <strong class="userinput"><code>atan2(0,0)</code></strong> vrací 0
namísto selhání.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Vypočítat funkci kosinus.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Vypočítat funkci hyperbolický kosinus.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce";
target="_top">Wikipedia</a> a <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-cot"></a>cot</span></dt><dd><pre class="synopsis">cot (x)</pre><p>Funkce kotangens.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Funkce hyperbolický kotangens.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce";
target="_top">Wikipedia</a> a <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-csc"></a>csc</span></dt><dd><pre class="synopsis">csc (x)</pre><p>Funkce kosekans.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Funkce hyperbolický kosekans.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Funkce sekans.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Funkce hyperbolický sekans.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Vypočítat funkci sinus.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Vypočítat funkci hyperbolický sinus.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce";
target="_top">Wikipedia</a> a <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-tan"></a>tan</span></dt><dd><pre class="synopsis">tan (x)</pre><p>Vypočítat funkci
tangens.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Funkce hyperbolický tangens.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce"; target="_top">Wikipedia</a> a
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s05.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s07.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Práce s
čísly </td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
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diff --git a/help/cs/html/ch11s07.html b/help/cs/html/ch11s07.html
index fa3bc24..b6c32fd 100644
--- a/help/cs/html/ch11s07.html
+++ b/help/cs/html/ch11s07.html
@@ -1,8 +1,71 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Teorie
čísel</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11.
Seznam funkcí GEL"><link rel="prev" href="ch11s06.html" title="Trigonometrie"><link rel="next"
href="ch11s08.html" title="Práce s maticemi"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Teorie čísel</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s06.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s08.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 c
lass="title" style="clear: both"><a name="genius-gel-function-list-number-theory"></a>Teorie
čísel</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>Jsou reálná celá čísla <code class="varname">a</code> a
<code class="varname">b</code> nesoudělná? Vrací <code class="constant">true</code> nebo <code
class="constant">false</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="https://cs.wikipedia.org/wiki/Nesoud%C4%9Bln%C3%A1_%C4%8D%C3%ADsla";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function
-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Vrátit <code class="varname">n</code>-té Bernoulliho číslo.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> (text je v angličtině) a <a class="ulink"
href="http://mathworld.wolfram.com/BernoulliNumber.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alternativní názvy: <code
class="function">CRT</code></p><p>Najít pomocí čínské věty o zbytcích <code class="varname">x</code>, které
řeší systém zadaný vektorem <code class="varname">a</code>, a zbytky prvků <code
class="varname">m</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/ChineseRe
mainderTheorem" target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/%C4%8C%C3%ADnsk%C3%A1_v%C4%9Bta_o_zbytc%C3%ADch";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Jsou-li dány dva rozklady, vrátit rozklad (faktorizaci)
součinu.</p><p>Viz <a class="link"
href="ch11s07.html#gel-function-Factorize">Factorize</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Převést vektor hodnot udávajících mocniny <code class="varname">b</code> na
číslo.</p></dd><dt><span class="term"><a name="gel-function-ConvertToBase"></a>Conve
rtToBase</span></dt><dd><pre class="synopsis">ConvertToBase (n,b)</pre><p>Převést číslo na vektor mocnin
prvků o základu <code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>Najít diskrétní logaritmus <code class="varname">n</code> o základu <code
class="varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code class="varname">q</code>, kde <code
class="varname">q</code> je prvočíslo, pomocí Silverova-Pohligova-Hellmanova algoritmu.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://planetmath.org/DiscreteLogarithm";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/DiscreteLogarithm.html"; target="_top">Mathworld</a> (text je v angličtině)
a <a class="ulink" href="http://cs.wikipedia.org/wiki/Diskr%C3%A9tn%C3%AD_logaritmus"; target="_top">Wikipedia
</a>.</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Zkontrolovat dělitelnost (zda <code class="varname">m</code> dělí
<code class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>Spočítat
Eulerovu funkci fí pro <code class="varname">n</code>, to je počet celých čísel mezi 1 a <code
class="varname">n</code>, relativně prvočíselných vůči <code class="varname">n</code>.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://planetmath.org/EulerPhifunction";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/TotientFunction.html"; target="_top">Mathworld</a> (text je v angličtině) a
<a class="ulink" href="http://cs.wikipedia.org/wiki/Eulerova_funkce";
target="_top">Wikipedia</a>.</p></dd><dt><spa
n class="term"><a name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre
class="synopsis">ExactDivision (n,d)</pre><p>Vrátit <strong class="userinput"><code>n/d</code></strong>, ale
jen pokud <code class="varname">d</code> dělí <code class="varname">n</code>. Pokud <code
class="varname">d</code> nedělí <code class="varname">n</code>, vrací funkce nesmysly. Pro velmi velká čísla
je to rychlejší než operace <strong class="userinput"><code>n/d</code></strong>, ale je to samozřejmě
použitelné jen v případě, kdy přesně víte, co dělíte.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize (n)</pre><p>Vrátit
rozklad (faktorizaci) čísla jako matici. První řádek jsou prvočísla v rozkladu (včetně 1) a druhý řádek jsou
mocnitelé. Takže například </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>Factorize(11*11*13)
</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Teorie
čísel</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11.
Seznam funkcí GEL"><link rel="prev" href="ch11s06.html" title="Trigonometrie"><link rel="next"
href="ch11s08.html" title="Práce s maticemi"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Teorie čísel</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s06.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s08.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 c
lass="title" style="clear: both"><a name="genius-gel-function-list-number-theory"></a>Teorie
čísel</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>Jsou reálná celá čísla <code class="varname">a</code> a
<code class="varname">b</code> nesoudělná? Vrací <code class="constant">true</code> nebo <code
class="constant">false</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="https://cs.wikipedia.org/wiki/Nesoud%C4%9Bln%C3%A1_%C4%8D%C3%ADsla";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function
-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Vrátit <code class="varname">n</code>-té Bernoulliho číslo.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alternativní názvy: <code
class="function">CRT</code></p><p>Najít pomocí čínské věty o zbytcích <code class="varname">x</code>, které
řeší systém zadaný vektorem <code class="varname">a</code>, a zbytky prvků <code
class="varname">m</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Jsou-li dány dva rozklady, vrátit rozklad (faktorizaci)
součinu.</p><p>Viz <a class="link"
href="ch11s07.html#gel-function-Factorize">Factorize</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Převést vektor hodnot udávajících mocniny <code class="varname">b</code> na
číslo.</p></dd><dt><span class="term"><a
name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre class="synopsis">ConvertToBase
(n,b)</pre><p>Převést číslo na vektor mocnin prvků o základu <code
class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>Najít diskrétní logaritmu
s <code class="varname">n</code> o základu <code class="varname">b</code> v F<sub>q</sub>, konečné grupě
řádu <code class="varname">q</code>, kde <code class="varname">q</code> je prvočíslo, pomocí
Silverova-Pohligova-Hellmanova algoritmu.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Zkontrolovat dělitelnost (zda <code class="varname">m</code> dělí
<code class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>Spočítat
Eulerovu funkci fí pro <code class="varname">n</code>, to je počet celých čísel mezi 1 a <code
class="varname">n</code>, relativně prvočíselných vůči <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Vrátit <strong class="userinput"><code>n/d</code></strong>, ale jen pokud <code
class="varname">d</code> dělí <code class="varname">n</code>. Pokud <code class="varname">d</code> nedělí
<code class="varname">n</code>, vrací funkce nesmysly. Pro velmi velká čísla je to rychlejší než operace
<strong class="userinput"><code>n/d</code></strong>, ale je to samozřejmě použitelné jen v případě, kdy
přesně víte, co dělíte.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize (n)</pre><p>Vrátit
rozklad (faktorizaci) čísla jako matici. První řádek jsou prvočísla v rozkladu (včetně 1) a druhý řádek jsou
mocnitelé. Takže například </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code
Factorize(11*11*13)</code></strong>
=
[1 11 13
- 1 2 1]</pre><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Faktorizace"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre class="synopsis">Factors
(n)</pre><p>Vrátit všechny činitele čísla <code class="varname">n</code> jako vektor. Součástí jsou i
neprvočíselní činitelé, což zahrnuje také 1 a přímo ono číslo. Takže například pro výpis všech dokonalých
čísel (to jsou taková, která jsou součtem všech svých činitelů) až do 1000 můžete udělat toto (je to však
značně neefektivní) </p><pre class="programlisting">for n=1 to 1000 do (
+ 1 2 1]</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>Vrátit všechny činitele čísla <code class="varname">n</code> jako
vektor. Součástí jsou i neprvočíselní činitelé, což zahrnuje také 1 a přímo ono číslo. Takže například pro
výpis všech dokonalých čísel (to jsou taková, která jsou součtem všech svých činitelů) až do 1000 můžete
udělat toto (je to však značně neefektivní) </p><pre class="programlisting">for n=1 to 1000 do (
if MatrixSum (Factors(n)) == 2*n then
print(n)
)
-</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,pokusy)</pre><p>Zkusit Fermatův rozklad <code
class="varname">n</code> na <strong class="userinput"><code>(t-s)*(t+s)</code></strong>. Pokud to je možné,
vrací <code class="varname">t</code> a <code class="varname">s</code> jako vektor, jinak vrací <code
class="constant">null</code>. Argument <code class="varname">pokusy</code> určuje počet pokusu, než se
výpočet vzdá.</p><p>Jedná se o docela dobrý rozklad za předpokladu, že je vaše číslo součinem dvou přibližně
stejně velkých čísel.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://en.wikipedia.org/wiki/Fermat_factorization"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre class="synops
is">FindPrimitiveElementMod (q)</pre><p>Najít první primitivní prvek v F<sub>q</sub>, konečné grupě řádu
<code class="varname">q</code>. Je samozřejmé, že <code class="varname">q</code> musí být
prvočíslo.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Najít náhodný primitivní prvek v F<sub>q</sub>,
konečné grupě řádu <code class="varname">q</code>. Je samozřejmé, že <code class="varname">q</code> musí být
prvočíslo.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Spočítat diskrétní logaritmus <code class="varname">n</code> o základu <code
class="varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code class="varname">q</code> (<code
class="varname">q</code> prvočíslo) pomocí
faktorizační báze <code class="varname">S</code>. <code class="varname">S</code> by měl být sloupec
prvočísel, pokud možno s druhým sloupcem předpočítaným pomocí <a class="link"
href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Provést přípravný krok výpočtu funkce <a
class="link" href="ch11s07.html#gel-function-IndexCalculus"><code class="function">IndexCalculus</code></a>
pro logaritmy o základu <code class="varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code
class="varname">q</code> (<code class="varname">q</code> prvočíslo), pro faktorizační bázi <code
class="varname">S</code> (kde <code class="varname">S</code> je sloupcový vektor prvočísel). Logaritmy budou
předp
očítány a vráceny v druhém sloupci.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven (n)</pre><p>Otestovat, zda
je celé číslo sudé.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Zjistit, jestli je kladné celé číslo <code
class="varname">p</code> Mersennovo prvočíslo. Tj. zda 2<sup>p</sup>-1 je prvočíslo. Provádí se to hledáním v
tabulce známých hodnot, která je relativně krátká. Viz také <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> a <a class="link"
href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a> (text je v
angličtině), <a class="ulink" href="ht
tp://mathworld.wolfram.com/MersennePrime.html" target="_top">Mathworld</a> (text je v angličtině), <a
class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Mersennovo_prvo%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Zjistit, jestli je racionální číslo <code class="varname">m</code> perfektní <code
class="varname">n</code>-tou mocninou . Viz také <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> a <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Otestovat, zda je
celé číslo liché.</p></dd><dt><span class="term"><a name="gel-function-IsPerf
ectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower (n)</pre><p>Zkontrolovat,
zda je celé číslo perfekntí mocninou a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>Zkontrolovat, zda je celé číslo perfektní druhou mocninou celého čísla. Číslo musí být přirozené
číslo. Záporná celá čísla samozřejmě perfektními druhými mocninami přirozených čísel být
nemohou.</p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>Testuje prvočíselnost celých čísel, pro čísla menší než 2.5e10 je
odpověď deterministická (tedy pokud je Riemannova hypotéza platná). Pro větší čísla závisí falešné kladné
odpovědi na <a class="link" href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMill
erRabinReps</code></a>. Což znamená, že pravděpodobnost nesprávné kladné odpovědi je ¼ umocněná na <code
class="function">IsPrimeMillerRabinReps</code>. Výchozí hodnota 22 dává pravděpodobnost zhruba
5.7e-14.</p><p>Když je vráceno <code class="constant">false</code>, můžete si být jisti, že se jedná o
složené číslo. Jestliže si potřebujete být absolutně jistí, že máte prvočíslo, můžete použít funkci <a
class="link" href="ch11s07.html#gel-function-MillerRabinTestSure"><code
class="function">MillerRabinTestSure</code></a>, ale může to trvat trochu déle.</p><p>Více informací najdete
v encyklopediích <a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> (text
je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo"; target="_top">Wikipedia</a>.
</p></dd><dt><span class="term"><a name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre
class="synopsis">IsPrimitiveMod (g,q)</pre><p>Zkontrolovat, zda je <code class="varname">g</code> primitivní
v F<sub>q</sub>, konečné grupě řádu <code class="varname">q</code>, kde <code class="varname">q</code> je
prvočíslo. Pokud <code class="varname">q</code> není prvočíslo, jsou výsledky nesmyslné.</p></dd><dt><span
class="term"><a
name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimitiveModWithPrimeFactors</span></dt><dd><pre
class="synopsis">IsPrimitiveModWithPrimeFactors (g,q,f)</pre><p>Zkontrolovat, zda je <code
class="varname">g</code> primitivní v F<sub>q</sub>, konečné grupě řádu <code class="varname">q</code>, kde
<code class="varname">q</code> je prvočíslo a <code class="varname">f</code> je vektor prvočíselných činitelů
<code class="varname">q</code>-1. Pokud <code class="varname">q</code> není prvočíslo, jsou výsle
dky nesmyslné.</p></dd><dt><span class="term"><a
name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre class="synopsis">IsPseudoprime
(n,b)</pre><p>Zda je <code class="varname">n</code> pseudoprvočíslo o základu <code class="varname">b</code>,
ale ne prvočíslo, tj. jestli <strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong>. Volá se
funkce <a class="link" href="ch11s07.html#gel-function-PseudoprimeTest"><code
class="function">PseudoprimeTest</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Zjistit, zda je <code class="varname">n</code> silné
pseudoprvočíslo o základu <code class="varname">b</code>, ale ne prvočíslo.</p></dd><dt><span class="term"><a
name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi (a,b)</pre><p>Alternativní
názvy: <code class="function">JacobiSymbol<
/code></p><p>Spočítat Jacobiho symbol (a/b) (b by mělo být liché).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Alternativní názvy: <code class="function">JacobiKroneckerSymbol</code></p><p>Spočítat Jacobiho
symbol (a/b) s Kroneckerovým rozšířením (a/2)=(2/a), když <code class="varname">a</code> je liché, nebo
(a/2)=0, když <code class="varname">a</code> je sudé.</p></dd><dt><span class="term"><a
name="gel-function-LeastAbsoluteResidue"></a>LeastAbsoluteResidue</span></dt><dd><pre
class="synopsis">LeastAbsoluteResidue (a,n)</pre><p>Vrátit zbytek <code class="varname">a</code> mod <code
class="varname">n</code> s nejmenší absolutní hodnotou (v intervalu -n/2 až n/2).</p></dd><dt><span
class="term"><a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre class="synopsis">Legendre
(a,p)</pre><p>Alternativní názvy: <code class="function
">LegendreSymbol</code></p><p>Spočítat Legendrův symbol (a/p).</p><p>Více informací najdete v encyklopediích
<a class="ulink" href="http://planetmath.org/LegendreSymbol"; target="_top">Planetmath</a> (text je v
angličtině), <a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Legendre%C5%AFv_symbol"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre
class="synopsis">LucasLehmer (p)</pre><p>Zjistit pomocí Lucasova-Lehmerova testu, zda je 2<sup>p</sup>-1
Mersennovo prvočíslo. Viz také <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> a <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://
en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test" target="_top">Wikipedia</a> (text je v
angličtině), <a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a> (text je
v angličtině) a <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> (text je v agličtině).</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Vrátit <code class="varname">n</code>-té Lucasovo číslo.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>
(text je v angličtině), <a class="ulink" href="http://planetmath.org/LucasNumbers";
target="_top">Planetmath</a> (text je v angličtině) a <a class="ulink"
href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span clas
s="term"><a name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Vrátit všechny maximální mocniny prvočísel v rozkladu
čísla.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>Vektor se známými exponenty Mersennových prvočísel, což je
seznam kladných celých čísel <code class="varname">p</code> takových, že 2<sup>p</sup>-1 je prvočíslo. Viz
také <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a> a
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>
(text je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/Mer
sennePrime.html" target="_top">Mathworld</a> (text je v angličtině), <a class="ulink"
href="http://www.mersenne.org/"; target="_top">GIMPS</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Mersennovo_prvo%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre class="synopsis">MillerRabinTest
(n,opak)</pre><p>Použít Millerův-Rabinův test prvočíselnosti na <code class="varname">n</code>, <code
class="varname">opak</code> udává kolikrát. Pravděpodobnost falešné kladné odpovědi je <strong
class="userinput"><code>(1/4)^opak</code></strong>. Pravděpodobně je obvykle lepší použít funkci <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a>, protože je
rychlejší a lepší u menších celých čísel.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planet
math.org/MillerRabinPrimeTest" target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html"; target="_top">Mathworld</a> (text
je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Miller%C5%AFv-Rabin%C5%AFv_test_prvo%C4%8D%C3%ADselnosti";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>Použít Millerův-Rabinův test prvočíselnosti na <code
class="varname">n</code> s tolika bázemi, že za předpokladu zobecněné Riemannovy hypotézy je výsledek
deterministický.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html"; t
arget="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Miller%C5%AFv-Rabin%C5%AFv_test_prvo%C4%8D%C3%ADselnosti";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre class="synopsis">ModInvert
(n,m)</pre><p>Vrátit převrácenou hodnotu n mod m.</p><p>Více informací najdete v encyklopedii <a
class="ulink" href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu (n)</pre><p>Vrátit
Möbiovu funkci μ vyhodnocenu na <code class="varname">n</code>. Což znamená, že vrátí 0 v případě, že <code
class="varname">n</code> není součin různých prvočísel, a <strong
class="userinput"><code>(-1)^k</code></strong> v případě, že je součin <code class="varname">k</code> r�
�zných prvočísel.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/MoebiusFunction"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/M%C3%B6biova_funkce";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre class="synopsis">NextPrime (n)</pre><p>Vrátit
nejmenší prvočíslo větší než <code class="varname">n</code>. Záporná prvočísla jsou považována za prvočísla,
takže předchozí prvočíslo můžete získat jako <strong
class="userinput"><code>-NextPrime(-n)</code></strong>.</p><p>Tato funkce používá funkci <code
class="function">mpz_nextprime</code> z knihovny GMP, která zase používá pravděpodobnostní Millerův-Rabinův
test (viz také <a class="
link" href="ch11s07.html#gel-function-MillerRabinTest"><code class="function">MillerRabinTest</code></a>).
Pravděpodobnost falešné kladné odpovědi není nastavitelná, ale je dostatečně malá pro praktické
účely.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synopsis">PadicValuation
(n,p)</pre><p>Vrátit p-adické ohodnocení (počet koncových nul v základu <code
class="varname">p</code>).</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://en.wikipedia.org/wiki/P-adic_order"; target="_t
op">Wikipedia</a> (text je v angličtině) a <a class="ulink" href="http://planetmath.org/PAdicValuation";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre class="synopsis">PowerMod
(a,b,m)</pre><p>Spočítat <strong class="userinput"><code>a^b mod m</code></strong>. <code
class="varname">b</code>-tá mocnina čísla <code class="varname">a</code> modulo <code
class="varname">m</code>. Tuto funkci není nutné používat, protože se automaticky použije v režimu modulární
aritmetiky. Z tohoto důvodu je <strong class="userinput"><code>a^b mod m</code></strong> stejně
rychlé.</p></dd><dt><span class="term"><a name="gel-function-Prime"></a>Prime</span></dt><dd><pre
class="synopsis">Prime (n)</pre><p>Alternativní názvy: <code class="function">prime</code></p><p>Vrátit <code
class="varname">n</code>-té prvočíslo (až do limitu).</p><p>Více informací najdete v encykl
opediích <a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> (text je v
angličtině), <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre
class="synopsis">PrimeFactors (n)</pre><p>Vrátit v podobě vektoru všechny prvočinitele čísla.</p><p>Více
informací najdete v encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/PrimeFactor.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">Pseudoprim
eTest (n,b)</pre><p>Test pseudoprvočíselnosti, vrací <code class="constant">true</code> když a jen když
<strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/Pseudoprime"; target="_top">Planetmath</a> (text
je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/Pseudoprime.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Pseudoprvo%C4%8D%C3%ADslo"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre
class="synopsis">RemoveFactor (n,m)</pre><p>Odstranit všechny instance činitele <code
class="varname">m</code> z čísla <code class="varname">n</code>. Prakticky to znamená, že je poděleno
nejvyšší mocninou čísla <code class="varname">m</code>, která je dělitelem <code class="varname">n</code>.</p>
<p>Více informací najdete v encyklopediích <a class="ulink" href="http://planetmath.org/Divisibility";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/Factor.html"; target="_top">Mathworld</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/D%C4%9Blitelnost";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-SilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Najít diskrétní logaritmus <code
class="varname">n</code> o základu <code class="varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code
class="varname">q</code>, kde <code class="varname">q</code> je prvočíslo, pomocí
Silverova-Pohligova-Hellmanova algoritmu, dané <code class="varname">f</code> je rozkladem <code
class="varname">q</code>-1.</p></dd><dt><span class="
term"><a name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Najít druhou odmocninu z <code class="varname">n</code> modulo <code class="varname">p</code>
(kde <code class="varname">p</code> je prvočíslo). Pokud není kvadratickým zbytkem, je vráceno
null.</p><p>Více informací najdete v encyklopedicíh <a class="ulink"
href="http://planetmath.org/QuadraticResidue"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://mathworld.wolfram.com/QuadraticResidue.html"; target="_top">Mathworld</a> (text je
v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Spustit silný test pseudoprvočíselnosti o základu <code
class="varname">b</code> na <code class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="h
ttp://planetmath.org/StrongPseudoprime" target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/StrongPseudoprime.html"; target="_top">Mathworld</a> (text je
v angličtině) a <a class="ulink" href="https://en.wikipedia.org/wiki/Strong_pseudoprime";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-gcd"></a>gcd</span></dt><dd><pre class="synopsis">gcd
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">GCD</code></p><p>Největší společný
dělitel celých čísel. V seznamu argumentů můžete uvést libovolný počet celých čísel, nebo je můžete zadat
jako vektor nebo matici celých čísel. Pokud zadáte více než jednu matici stejné velikosti, bude největší
společný dělitel určen prvek po prvku.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmat
h</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/GreatestCommonDivisor.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Nejv%C4%9Bt%C5%A1%C3%AD_spole%C4%8Dn%C3%BD_d%C4%9Blitel";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-lcm"></a>lcm</span></dt><dd><pre class="synopsis">lcm
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">LCM</code></p><p>Nejmenší společný
násobek celých čísel. V seznamu argumentů můžete uvést libovolný počet celých čísel, nebo je můžete zadat
jako vektor nebo matici celých čísel. Pokud zadáte více než jednu matici stejné velikosti, bude nejmenší
společný násobek určen prvek po prvku.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/LeastCommonMultiple"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" hr
ef="http://mathworld.wolfram.com/LeastCommonMultiple.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Nejmen%C5%A1%C3%AD_spole%C4%8Dn%C3%BD_n%C3%A1sobek";
target="_top">Wikipedia</a>.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s06.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
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valign="top"> Práce s maticemi</td></tr></table></div></body></html>
+</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,pokusy)</pre><p>Zkusit Fermatův rozklad <code
class="varname">n</code> na <strong class="userinput"><code>(t-s)*(t+s)</code></strong>. Pokud to je možné,
vrací <code class="varname">t</code> a <code class="varname">s</code> jako vektor, jinak vrací <code
class="constant">null</code>. Argument <code class="varname">pokusy</code> určuje počet pokusu, než se
výpočet vzdá.</p><p>Jedná se o docela dobrý rozklad za předpokladu, že je vaše číslo součinem dvou přibližně
stejně velkých čísel.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Najít první primitivní prvek v F<sub>q</sub>, konečné
grupě řádu <code class="varname">q</code>. Je samozřejmé, že <code class="varname">q</code> musí být
prvočíslo.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Najít náhodný primitivní prvek v F<sub>q</sub>,
konečné grupě řádu <code class="varname">q</code>. Je samozřejmé, že <code class="varname">q</code> musí být
prvočíslo.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Spočítat diskrétní logaritmus <code class="varname">n</code> o základu <code class="
varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code class="varname">q</code> (<code
class="varname">q</code> prvočíslo) pomocí faktorizační báze <code class="varname">S</code>. <code
class="varname">S</code> by měl být sloupec prvočísel, pokud možno s druhým sloupcem předpočítaným pomocí <a
class="link" href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Provést přípravný krok výpočtu funkce <a
class="link" href="ch11s07.html#gel-function-IndexCalculus"><code class="function">IndexCalculus</code></a>
pro logaritmy o základu <code class="varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code
class="varname">q</code> (<code class="varname">q</code> prvočíslo), pro fa
ktorizační bázi <code class="varname">S</code> (kde <code class="varname">S</code> je sloupcový vektor
prvočísel). Logaritmy budou předpočítány a vráceny v druhém sloupci.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven (n)</pre><p>Otestovat, zda
je celé číslo sudé.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Zjistit, jestli je kladné celé číslo <code
class="varname">p</code> Mersennovo prvočíslo. Tj. zda 2<sup>p</sup>-1 je prvočíslo. Provádí se to hledáním v
tabulce známých hodnot, která je relativně krátká. Viz také <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> a <a class="link"
href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
+ for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Zjistit, jestli je racionální číslo <code class="varname">m</code> perfektní <code
class="varname">n</code>-tou mocninou . Viz také <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> a <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Otestovat, zda je
celé číslo liché.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Zkontrolovat, zda je celé číslo perfekntí mocninou a<sup>b</sup>.</p></dd><dt><span
class="term"><a name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre
class="synopsis">IsPerfectSquare (n)<
/pre><p>
+ Check an integer for being a perfect square of an integer. The number must
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
+ </p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>Testuje prvočíselnost celých čísel, pro čísla menší než 2.5e10 je
odpověď deterministická (tedy pokud je Riemannova hypotéza platná). Pro větší čísla závisí falešné kladné
odpovědi na <a class="link" href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. Což znamená, že pravděpodobnost nesprávné kladné odpovědi
je ¼ umocněná na <code class="function">IsPrimeMillerRabinReps</code>. Výchozí hodnota 22 dává
pravděpodobnost zhruba 5.7e-14.</p><p>Když je vráceno <code class="constant">false</code>, můžete si být
jisti, že se jedná o složené číslo. Jestliže si potřebujete být absolutně jistí, že máte prvočíslo, můžete
použít funkci <a class="link" href="ch11s07.html#gel-function-MillerRabinTestSure"><code class="function">Mil
lerRabinTestSure</code></a>, ale může to trvat trochu déle.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> (text je v angličtině),
<a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre class="synopsis">IsPrimitiveMod
(g,q)</pre><p>Zkontrolovat, zda je <code class="varname">g</code> primitivní v F<sub>q</sub>, konečné grupě
řádu <code class="varname">q</code>, kde <code class="varname">q</code> je prvočíslo. Pokud <code
class="varname">q</code> není prvočíslo, jsou výsledky nesmyslné.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimitiveModWithPrime
Factors</span></dt><dd><pre class="synopsis">IsPrimitiveModWithPrimeFactors (g,q,f)</pre><p>Zkontrolovat,
zda je <code class="varname">g</code> primitivní v F<sub>q</sub>, konečné grupě řádu <code
class="varname">q</code>, kde <code class="varname">q</code> je prvočíslo a <code class="varname">f</code> je
vektor prvočíselných činitelů <code class="varname">q</code>-1. Pokud <code class="varname">q</code> není
prvočíslo, jsou výsledky nesmyslné.</p></dd><dt><span class="term"><a
name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre class="synopsis">IsPseudoprime
(n,b)</pre><p>Zda je <code class="varname">n</code> pseudoprvočíslo o základu <code class="varname">b</code>,
ale ne prvočíslo, tj. jestli <strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong>. Volá se
funkce <a class="link" href="ch11s07.html#gel-function-PseudoprimeTest"><code
class="function">PseudoprimeTest</code></a>.</p></dd><dt><span class="term"><a name="g
el-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Zjistit, zda je <code class="varname">n</code> silné
pseudoprvočíslo o základu <code class="varname">b</code>, ale ne prvočíslo.</p></dd><dt><span class="term"><a
name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi (a,b)</pre><p>Alternativní
názvy: <code class="function">JacobiSymbol</code></p><p>Spočítat Jacobiho symbol (a/b) (b by mělo být
liché).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Alternativní názvy: <code class="function">JacobiKroneckerSymbol</code></p><p>Spočítat Jacobiho
symbol (a/b) s Kroneckerovým rozšířením (a/2)=(2/a), když <code class="varname">a</code> je liché, nebo
(a/2)=0, když <code class="varname">a</code> je sudé.</p></dd><dt><span class="term"><a name="ge
l-function-LeastAbsoluteResidue"></a>LeastAbsoluteResidue</span></dt><dd><pre
class="synopsis">LeastAbsoluteResidue (a,n)</pre><p>Vrátit zbytek <code class="varname">a</code> mod <code
class="varname">n</code> s nejmenší absolutní hodnotou (v intervalu -n/2 až n/2).</p></dd><dt><span
class="term"><a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre class="synopsis">Legendre
(a,p)</pre><p>Alternativní názvy: <code class="function">LegendreSymbol</code></p><p>Spočítat Legendrův
symbol (a/p).</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/LegendreSymbol"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="https://cs.wikipedia.org/wiki/Legendre%C5%AFv_symbol";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function-LucasLehmer"></a>Luc
asLehmer</span></dt><dd><pre class="synopsis">LucasLehmer (p)</pre><p>Zjistit pomocí Lucasova-Lehmerova
testu, zda je 2<sup>p</sup>-1 Mersennovo prvočíslo. Viz také <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> a <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Vrátit <code class="varname">n</code>-té Lucasovo číslo.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Vrátit všechny maximální mocniny prvočísel v rozkladu
čísla.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>Vektor se známými exponenty Mersennových prvočísel, což je
seznam kladných celých čísel <code class="varname">p</code> takových, že 2<sup>p</sup>-1 je prvočíslo. Viz
také <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a> a
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
+ for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre class="synopsis">MillerRabinTest
(n,opak)</pre><p>Použít Millerův-Rabinův test prvočíselnosti na <code class="varname">n</code>, <code
class="varname">opak</code> udává kolikrát. Pravděpodobnost falešné kladné odpovědi je <strong
class="userinput"><code>(1/4)^opak</code></strong>. Pravděpodobně je obvykle lepší použít funkci <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a>, protože je
rychlejší a lepší u menších celých čísel.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>Použít Millerův-Rabinův test prvočíselnosti na <code
class="varname">n</code> s tolika bázemi, že za předpokladu zobecněné Riemannovy hypotézy je výsledek
deterministický.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
+ <a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Vrátit převrácenou hodnotu n mod m.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="http://mathworld.wolfram.com/ModularInverse.html";
target="_top">Mathworld</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu (n)</pre><p>Vrátit
Möbiovu funkci μ vyhodnocenu na <code class="varname">n</code>. Což znamená, že vrátí 0 v případě, že <code
class="varname">n</code> není součin různých prvočísel, a <strong
class="userinput"><code>(-1)^k</code></strong> v případě, že je součin <code class="varname">k</code> různých
prvočísel.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/MoebiusFunction"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html"; target="_top">Mathworld</a> (text je
v angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/M%C3%B6biova_funkce";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre class="synopsis">NextPrime (n)</pre><p>Vrátit
nejmenší prvočíslo větší než <code class="varname">n</code>. Záporná prvočísla jsou považována za prvočísla,
takže předchozí prvočíslo můžete získat jako <strong
class="userinput"><code>-NextPrime(-n)</code></strong>.</p><p>Tato funkce používá funkci <code
class="function">mpz_nextprime</code> z knihovny GMP, která zase používá pravděpodobnostní Millerův-Rabinův
test (viz také <a class="link" href="ch11s07.html#gel-function-MillerRabinTest"><code
class="function">MillerRabinTest</code></a>). Pravděpodobnost falešné kladné odpovědi není nastavitelná, ale
je dostateč
ně malá pro praktické účely.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synopsis">PadicValuation
(n,p)</pre><p>Vrátit p-adické ohodnocení (počet koncových nul v základu <code
class="varname">p</code>).</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://en.wikipedia.org/wiki/P-adic_order"; target="_top">Wikipedia</a> (text je v angličtině) a <a
class="ulink" href="http://planetmath.org/PAdicValuation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="ter
m"><a name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre class="synopsis">PowerMod
(a,b,m)</pre><p>Spočítat <strong class="userinput"><code>a^b mod m</code></strong>. <code
class="varname">b</code>-tá mocnina čísla <code class="varname">a</code> modulo <code
class="varname">m</code>. Tuto funkci není nutné používat, protože se automaticky použije v režimu modulární
aritmetiky. Z tohoto důvodu je <strong class="userinput"><code>a^b mod m</code></strong> stejně
rychlé.</p></dd><dt><span class="term"><a name="gel-function-Prime"></a>Prime</span></dt><dd><pre
class="synopsis">Prime (n)</pre><p>Alternativní názvy: <code class="function">prime</code></p><p>Vrátit <code
class="varname">n</code>-té prvočíslo (až do limitu).</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> (text je v angličtině),
<a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.ht
ml" target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADslo"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre
class="synopsis">PrimeFactors (n)</pre><p>Vrátit v podobě vektoru všechny prvočinitele čísla.</p><p>Více
informací najdete v encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/PrimeFactor.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">PseudoprimeTest
(n,b)</pre><p>Test pseudoprvočíselnosti, vrací <code class="constant">true</code> když a jen když <strong
class="userinput"><code>b^(n-1) == 1 mod n</code></strong>.</p><p>Více
informací najdete v encyklopediích <a class="ulink" href="http://planetmath.org/Pseudoprime";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/Pseudoprime.html"; target="_top">Mathworld</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Pseudoprvo%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre class="synopsis">RemoveFactor
(n,m)</pre><p>Odstranit všechny instance činitele <code class="varname">m</code> z čísla <code
class="varname">n</code>. Prakticky to znamená, že je poděleno nejvyšší mocninou čísla <code
class="varname">m</code>, která je dělitelem <code class="varname">n</code>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/Divisibility"; target="_top">Planetmath</a> (text
je v angličtině), <a class="ulink" href="http:/
/mathworld.wolfram.com/Factor.html" target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/D%C4%9Blitelnost"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a
name="gel-function-SilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Najít diskrétní logaritmus <code
class="varname">n</code> o základu <code class="varname">b</code> v F<sub>q</sub>, konečné grupě řádu <code
class="varname">q</code>, kde <code class="varname">q</code> je prvočíslo, pomocí
Silverova-Pohligova-Hellmanova algoritmu, dané <code class="varname">f</code> je rozkladem <code
class="varname">q</code>-1.</p></dd><dt><span class="term"><a
name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Najít druhou odmocninu z <code class="varname">n</code> modulo <c
ode class="varname">p</code> (kde <code class="varname">p</code> je prvočíslo). Pokud není kvadratickým
zbytkem, je vráceno null.</p><p>Více informací najdete v encyklopedicíh <a class="ulink"
href="http://planetmath.org/QuadraticResidue"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://mathworld.wolfram.com/QuadraticResidue.html"; target="_top">Mathworld</a> (text je
v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Spustit silný test pseudoprvočíselnosti o základu <code
class="varname">b</code> na <code class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/StrongPseudoprime"; target="_top">Planetmath</a> (text je v
angličtině), <a class="ulink" href="http://mathworld.wolfram.com/StrongPseudoprime.html"; target="_top">Math
world</a> (text je v angličtině) a <a class="ulink" href="https://en.wikipedia.org/wiki/Strong_pseudoprime";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-gcd"></a>gcd</span></dt><dd><pre class="synopsis">gcd
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">GCD</code></p><p>Největší společný
dělitel celých čísel. V seznamu argumentů můžete uvést libovolný počet celých čísel, nebo je můžete zadat
jako vektor nebo matici celých čísel. Pokud zadáte více než jednu matici stejné velikosti, bude největší
společný dělitel určen prvek po prvku.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/GreatestCommonDivisor.html"; target="_top">Mathworld</a>
(text je v angličtině) a <a class="ulink" href=
"https://cs.wikipedia.org/wiki/Nejv%C4%9Bt%C5%A1%C3%AD_spole%C4%8Dn%C3%BD_d%C4%9Blitel";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-lcm"></a>lcm</span></dt><dd><pre class="synopsis">lcm
(a,argumenty...)</pre><p>Alternativní názvy: <code class="function">LCM</code></p><p>Nejmenší společný
násobek celých čísel. V seznamu argumentů můžete uvést libovolný počet celých čísel, nebo je můžete zadat
jako vektor nebo matici celých čísel. Pokud zadáte více než jednu matici stejné velikosti, bude nejmenší
společný násobek určen prvek po prvku.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/LeastCommonMultiple"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/LeastCommonMultiple.html"; target="_top">Mathworld</a> (text
je v angličtině) a <a class="ulink" href="https://cs.wikipedia.org/wiki/Nejmen%C5%A1%C3%AD_spo
le%C4%8Dn%C3%BD_n%C3%A1sobek" target="_top">Wikipedia</a>.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
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-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Práce s
maticemi</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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Seznam funkcí GEL"><link rel="prev" href="ch11s07.html" title="Teorie čísel"><link rel="next"
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header"><tr><th colspan="3" align="center">Práce s maticemi</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s07.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s09.html">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><d
iv><h2 class="title" style="clear: both"><a name="genius-gel-function-list-matrix"></a>Práce s
maticemi</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,fce)</pre><p>Použít funkci na všechny prvky matice a vrátit matici výsledků.</p></dd><dt><span
class="term"><a name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,fce)</pre><p>Použít funkci na všechny prvky 2 matic (nebo 1 hodnoty a
1 matice) a vrátit matici výsledků.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Vrátit
sloupce matice jako vodorovný vektor.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre class="synopsis">Compleme
ntSubmatrix (m,r,c)</pre><p>Odstranit sloupec (či slupce) a řádek (či řádky) z matice.</p></dd><dt><span
class="term"><a name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre
class="synopsis">CompoundMatrix (k,A)</pre><p>Spočítat <code class="varname">k</code>-tou složenou matici
matice A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>Spočítat počet nulových sloupců v matici. Například jakmile
zredukujete sloupce matice, můžete to využít k nalezení nulovosti. Viz <a class="link"
href="ch11s09.html#gel-function-cref"><code class="function">cref</code></a> a <a class="link"
href="ch11s09.html#gel-function-Nullity"><code class="function">Nullity</code></a>.</p></dd><dt><span
class="term"><a name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre
class="synopsis">DeleteColumn (M,sloupec)</pre><p>Smazat sloupec mati
ce.</p></dd><dt><span class="term"><a name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre
class="synopsis">DeleteRow (M,radek)</pre><p>Smazat řádek matice.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Získat diagonální prvky matice jako sloupcový vektor.</p><p>Více informací najdete v encyklopedii
<a class="ulink" href="http://cs.wikipedia.org/wiki/Diagon%C3%A1ln%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Získat skalární součin dvou vektorů. Vektory musí mít stejnou velikost. Nepřijímají se
konjugované vektory, protože jde o bilineární formu, i když pracuje i s komplexními čísly. Jedná se o
bilineární skalární součin, ne půldruhý lineární (seskvilineární). Pro ten slouží funkce <a class="lin
k" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a></p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/DotProduct"; target="_top">Planetmath</a> (text je
v angličtině) a <a class="ulink" href="https://cs.wikipedia.org/wiki/Skal%C3%A1rn%C3%AD_sou%C4%8Din";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre class="synopsis">ExpandMatrix
(M)</pre><p>Rozšířit matici, stejně když zadáte matici bez uvozovky. Takto se rozbalí do bloku libovolná
interní matice. Je to způsob, jak sestrojit matice z jiných menších a normálně je to prováděno na vstupu
automaticky, s výjimkou kdy je matice zadána s uvozovkou.</p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre
class="synopsis">HermitianProduct (u,v)</pre><p>Alternativní názvy: <code class="function">InnerProduct<
/code></p><p>Získat hermitovský součin dvou vektorů. Vektory musí mít stejnou velikost. Jedná se o
polybilineární formu používající jednotkovou matici.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://mathworld.wolfram.com/HermitianInnerProduct.html"; target="_top">Mathworld</a>
(text je v angličtině) a <a class="ulink" href="https://en.wikipedia.org/wiki/Sesquilinear_form";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-I"></a>I</span></dt><dd><pre class="synopsis">I (n)</pre><p>Alternativní názvy: <code
class="function">eye</code></p><p>Vrátit jednotkovou matici zadané velikosti, tj. <code
class="varname">n</code> krát <code class="varname">n</code>. Pokud je <code class="varname">n</code> rovno
0, vrátí <code class="constant">null</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/IdentityMatrix"; target="_top">Pla
netmath</a> (text je v angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Jednotkov%C3%A1_matice"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre
class="synopsis">IndexComplement (vektor,mvelikost)</pre><p>Vrátit doplňkový index vektoru indexů. Vše je s
jednou bází. Například pro vektor <strong class="userinput"><code>[2,3]</code></strong> a velikost <strong
class="userinput"><code>5</code></strong> dostaneme <strong class="userinput"><code>[1,4,5]</code></strong>.
Pokud je <code class="varname">mvelikost</code> rovna 0, vrací vždy <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Je
matice diagonální?</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetm
ath</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Diagon%C3%A1ln%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Zkontrolovat, zda je matice jednotková. Pokud matice není čtvercová, tak automaticky vrátí <code
class="varname">false</code>. Funguje i pro čísla, v kterémžto případě je to stejné jako <strong
class="userinput"><code>x==1</code></strong>. Pokud je argument <code class="varname">x</code> roven <code
class="constant">null</code> (což můžeme považovat za matici 0 krát 0), nezpůsobí to chybu a vrátí <code
class="constant">false</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Jde o dolní trojúhelníkovou matici? To je taková, která má
všechny prvky
nad diagonálou nulové.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Zkontrolovat, zda je matice maticí celých (nekomplexních) čísel.</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Zkontrolovat, zda je matice nezáporná, tj. zda je každý z
prvků nezáporný. Nepleťte si pozitivní matice s pozitivně definitními maticemi.</p><p>Více informací najdete
v encyklopedii <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Zkontrolovat, zda je matice pozitivní, tj. zda je každý z prvků
kladný (a tudíž reálný)
. Především není žádný prvek 0. Nepleťte si positivní matice s pozitivně definitními maticemi.</p><p>Více
informací najdete v encyklopedii <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Zkontrolovat, zda je matice maticí z racionálních
(nekomplexních) čísel.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Zkontrolovat, zda je matice složená z reálných (na komplexních) čísel.</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Zkontrolovat, zda je matice čtvercová, tj. šířka je stejná jako
výška.</p></dd><dt><span clas
s="term"><a name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Jde o horní trojúhelníkovou matici? To je taková, která má
všechny prvky pod diagonálou nulové.</p></dd><dt><span class="term"><a
name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Zkontrolovat, zda se matice skládá pouze z čísel. Mnoho interních funkcí provádí tuto kontrolu.
Hodnoty mohou být libovolná čísla včetně komplexních.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Je
argument vodorovný nebo svislý vektor? Genius nerozlišuje mezi maticí a vektorem, vektor je prostě jen matice
1 krát <code class="varname">n</code> nebo <code class="varname">n</code> krát 1.</p></dd><dt><span
class="term"><a name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="sy
nopsis">IsZero (x)</pre><p>Zkontrolovat, zda se matice skládá jen z nul. Funguje to i pro čísla, kdy je to
ekvivalentní výrazu <strong class="userinput"><code>x==0</code></strong>. Když je <code
class="varname">x</code> rovno <code class="constant">null</code> (můžeme to považovat za matici 0 krát 0),
nezpůsobí to žádnou chybu, ale vrátí se <code class="constant">true</code>, protože podmínka je
prázdná.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Vrátit kopii matice <code class="varname">M</code> se všemi prvky nad diagonálou nastavenými na
nulu.</p></dd><dt><span class="term"><a name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre
class="synopsis">MakeDiagonal (v,argument...)</pre><p>Alternativní názvy: <code
class="function">diag</code></p><p>Vytvořit diagonální matici z vektoru. Případně můžete hodnoty, které
se mají umístit na diagonálu, zadat jako jednotlivé parametry. Takže <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> je to stejné jako <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> (text je v angličtině)
nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Diagon%C3%A1ln%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Vytvořit sloupcový vektor z matice poskládáním sloupců na sebe. Pokud je předáno <code
class="constant">null</code>, vrátí <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>Spočítat součin v
šech prvků matice nebo vektoru. To znamená, že se vynásobí všechny prvky a vrátí se číslo, které je násobkem
všech těchto prvků.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre class="synopsis">MatrixSum
(A)</pre><p>Spočítat součet všech prvků matice nebo vektoru. To znamená, že se sečtou všechny prvky a vrátí
se číslo, které je součtem všech těchto prvků.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Spočítat součet druhých mocnin všech prvků matice nebo
vektoru.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Vrátit řádkový vektor s indexy nenulových sloupců v matici <code
class="varname">M</code>.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a na
me="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Vrátit řádkový vektor s indexy nenulových prvků ve vektoru <code
class="varname">v</code>.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Získat vnější součin dvou vektorů. Takže, když dejme tomu jsou <code class="varname">u</code> a
<code class="varname">v</code> svislé vektory, pak vnější součin je <strong class="userinput"><code>v *
u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Převrátit pořadí prvků ve vektoru. Pokud je předáno <code class="constant">null</code>, tak vrací
<code class="constant">null</code>.</p></dd><dt><span class="term"><a name="gel-function-RowSum"></a>Ro
wSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Vypočítat součet každého řádku v matici a
vrátit svislý vektor s výsledkem.</p></dd><dt><span class="term"><a
name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre class="synopsis">RowSumSquares
(m)</pre><p>Vypočítat součet druhých mocnin každého řádku v matici a vrátit svislý vektor s
výsledkem.</p></dd><dt><span class="term"><a name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre
class="synopsis">RowsOf (M)</pre><p>Získat řádky matice jako svislý vektor. Každý z prvků vektoru je
vodorovný vektor, který odpovídá řádku matice <code class="varname">M</code>. Tato funkce je užitečná, když
chcete ve smyčce procházet řádky matice. Například takto: <strong class="userinput"><code>for r in RowsOf(M)
do
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iv><h2 class="title" style="clear: both"><a name="genius-gel-function-list-matrix"></a>Práce s
maticemi</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,fce)</pre><p>Použít funkci na všechny prvky matice a vrátit matici výsledků.</p></dd><dt><span
class="term"><a name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,fce)</pre><p>Použít funkci na všechny prvky 2 matic (nebo 1 hodnoty a
1 matice) a vrátit matici výsledků.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Vrátit
sloupce matice jako vodorovný vektor.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre class="synopsis">Compleme
ntSubmatrix (m,r,c)</pre><p>Odstranit sloupec (či slupce) a řádek (či řádky) z matice.</p></dd><dt><span
class="term"><a name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre
class="synopsis">CompoundMatrix (k,A)</pre><p>Spočítat <code class="varname">k</code>-tou složenou matici
matice A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
+ the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
+ and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,sloupec)</pre><p>Smazat sloupec matice.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,radek)</pre><p>Smazat řádek matice.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Získat diagonální prvky matice jako sloupcový vektor.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Získat skalární součin dvou vektorů. Vektory musí mít stejnou velikost. Nepřijímají se
konjugované vektory, protože jde o bilineární formu, i když pracuje i s komplexními čísly. Jedná se o
bilineární skalární součin, ne půldruhý lineární (seskvilineární). Pro ten slouží funkce <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a></p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/DotProduct"; target="_top">Planetmath</a> (text je
v angličtině) a <a class="ulink" href="https://cs.wikipedia.org/wiki/Skal%C3%A1rn%C3%AD_sou%C4%8Din";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre class="synopsis">ExpandMatrix
(M)</pre><p>Rozšířit matic
i, stejně když zadáte matici bez uvozovky. Takto se rozbalí do bloku libovolná interní matice. Je to způsob,
jak sestrojit matice z jiných menších a normálně je to prováděno na vstupu automaticky, s výjimkou kdy je
matice zadána s uvozovkou.</p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre
class="synopsis">HermitianProduct (u,v)</pre><p>Alternativní názvy: <code
class="function">InnerProduct</code></p><p>Získat hermitovský součin dvou vektorů. Vektory musí mít stejnou
velikost. Jedná se o polybilineární formu používající jednotkovou matici.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/HermitianInnerProduct.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="https://en.wikipedia.org/wiki/Sesquilinear_form"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name
="gel-function-I"></a>I</span></dt><dd><pre class="synopsis">I (n)</pre><p>Alternativní názvy: <code
class="function">eye</code></p><p>Vrátit jednotkovou matici zadané velikosti, tj. <code
class="varname">n</code> krát <code class="varname">n</code>. Pokud je <code class="varname">n</code> rovno
0, vrátí <code class="constant">null</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="https://cs.wikipedia.org/wiki/Jednotkov%C3%A1_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vektor,mvelikost)</pre><p>Vrátit doplňkový index vektoru indexů. Vše je s jednou bází. Například pro vektor
<strong class="userinput"><code>[2,3]</code></strong> a velikost <strong class="userinput"><code>5</code></
strong> dostaneme <strong class="userinput"><code>[1,4,5]</code></strong>. Pokud je <code
class="varname">mvelikost</code> rovna 0, vrací vždy <code class="constant">null</code>.</p></dd><dt><span
class="term"><a name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal
(M)</pre><p>Je matice diagonální?</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Zkontrolovat, zda je matice jednotková. Pokud matice není čtvercová, tak automaticky vrátí <code
class="varname">false</code>. Funguje i pro čísla, v kterémžto případě je to stejné jako <strong
class="userinput"><code>x==1</code></strong>. Pokud je argument <code class="varname">x</code> roven <code
class="constant">null</code> (což můžeme považovat za matici 0 krát 0), nezpůsobí to chybu a vrátí <code
class="constant">false</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Jde o dolní trojúhelníkovou matici? To je taková, která má
všechny prvky nad diagonálou nulové.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="sy
nopsis">IsMatrixInteger (M)</pre><p>Zkontrolovat, zda je matice maticí celých (nekomplexních)
čísel.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Zkontrolovat, zda je matice nezáporná, tj. zda je každý z
prvků nezáporný. Nepleťte si pozitivní matice s pozitivně definitními maticemi.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Zkontrolovat, zda je matice pozitivní, tj. zda je každý z prvků
kladný (a tudíž reálný). Především není žádný prvek 0. Nepleťte si positivní matice s pozitivně definitními
maticemi.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Zkontrolovat, zda je matice maticí z racionálních
(nekomplexních) čísel.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Zkontrolovat, zda je matice složená z reálných (na komplexních) čísel.</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Zkontrolovat, zda je matice čtvercová, tj. šířka je stejná jako
výška.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Jde o horní trojúhelníkovou matici? To je taková, která má
všechny prvky pod diagonálou nulové.</p></dd><dt><span class="te
rm"><a name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Zkontrolovat, zda se matice skládá pouze z čísel. Mnoho interních funkcí provádí tuto kontrolu.
Hodnoty mohou být libovolná čísla včetně komplexních.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Je
argument vodorovný nebo svislý vektor? Genius nerozlišuje mezi maticí a vektorem, vektor je prostě jen matice
1 krát <code class="varname">n</code> nebo <code class="varname">n</code> krát 1.</p></dd><dt><span
class="term"><a name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero
(x)</pre><p>Zkontrolovat, zda se matice skládá jen z nul. Funguje to i pro čísla, kdy je to ekvivalentní
výrazu <strong class="userinput"><code>x==0</code></strong>. Když je <code class="varname">x</code> rovno
<code class="constant">null</code> (m�
�žeme to považovat za matici 0 krát 0), nezpůsobí to žádnou chybu, ale vrátí se <code
class="constant">true</code>, protože podmínka je prázdná.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Vrátit kopii matice <code class="varname">M</code> se všemi prvky nad diagonálou nastavenými na
nulu.</p></dd><dt><span class="term"><a name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre
class="synopsis">MakeDiagonal (v,argument...)</pre><p>Alternativní názvy: <code
class="function">diag</code></p><p>Vytvořit diagonální matici z vektoru. Případně můžete hodnoty, které se
mají umístit na diagonálu, zadat jako jednotlivé parametry. Takže <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> je to stejné jako <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Vytvořit sloupcový vektor z matice poskládáním sloupců na sebe. Pokud je předáno <code
class="constant">null</code>, vrátí <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>Spočítat součin všech prvků matice nebo vektoru. To znamená, že se vynásobí všechny prvky a vrátí
se číslo, které je násobkem všech těchto prvků.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre class="synopsis">MatrixSum
(A)</pre><p>Spočítat součet všech prvků matice nebo vektoru. To znamená, že se sečtou všechny prvky a vrátí
se číslo, které je součtem všech těchto prvků.</p></dd><dt><span class="term"><a name="gel-function-Matr
ixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre class="synopsis">MatrixSumSquares
(A)</pre><p>Spočítat součet druhých mocnin všech prvků matice nebo vektoru.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Vrátit řádkový vektor s indexy nenulových sloupců v matici <code
class="varname">M</code>.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Vrátit řádkový vektor s indexy nenulových prvků ve vektoru <code
class="varname">v</code>.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Získat vnější součin dvou vektorů. Takže, když dejme tomu jsou <code class="varname">u</code> a
<code class="varname">v</code> svislé vektory, pak vnější součin je <strong class="userinput"><code>v *
u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Převrátit pořadí prvků ve vektoru. Pokud je předáno <code class="constant">null</code>, tak vrací
<code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Vypočítat
součet každého řádku v matici a vrátit svislý vektor s výsledkem.</p></dd><dt><span class="term"><a
name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre class="synopsis">RowSumSquares
(m)</pre><p>Vypočítat součet druhých mocnin každého řádku v matici a vrátit svislý vektor s
výsledkem.</p></dd><dt><span class="term"><a name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre
class="syn
opsis">RowsOf (M)</pre><p>Získat řádky matice jako svislý vektor. Každý z prvků vektoru je vodorovný vektor,
který odpovídá řádku matice <code class="varname">M</code>. Tato funkce je užitečná, když chcete ve smyčce
procházet řádky matice. Například takto: <strong class="userinput"><code>for r in RowsOf(M) do
neco(r)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-SetMatrixSize"></a>SetMatrixSize</span></dt><dd><pre class="synopsis">SetMatrixSize
(M,radku,sloupcu)</pre><p>Vytvořit novou matici zadané velikosti z jiné staré. To znamená, že nová matice
bude vrácena jako kopie té staré. Prvky, které přebývají, jsou odříznuty a volné místo je vyplněno nulami.
Pokud je argument <code class="varname">radku</code> nebo <code class="varname">sloupcu</code> roven nule, je
vráceno <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-ShuffleVector"></a>ShuffleVector</span></dt><dd><pre class="synopsis">ShuffleVector
(v)</pre><p>Zamíchat pořadí prvků ve vektoru. Pokud je předáno <code class="constant">null</code>, tak vrací
<code class="constant">null</code>.</p><p>Verze 1.0.13 a novější.</p></dd><dt><span class="term"><a
name="gel-function-SortVector"></a>SortVector</span></dt><dd><pre class="synopsis">S
ortVector (v)</pre><p>Seřadit prvky vektoru ve vzestupném pořadí.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroColumns"></a>StripZeroColumns</span></dt><dd><pre
class="synopsis">StripZeroColumns (M)</pre><p>Odstranit všechny čistě nulové sloupce matice <code
class="varname">M</code>.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroRows"></a>StripZeroRows</span></dt><dd><pre class="synopsis">StripZeroRows
(M)</pre><p>Odstranit všechny čistě nulové řádky matice <code class="varname">M</code>.</p></dd><dt><span
class="term"><a name="gel-function-Submatrix"></a>Submatrix</span></dt><dd><pre class="synopsis">Submatrix
(m,r,s)</pre><p>Vrátit sloupec (či sloupce) a řádek (či řádky) z matice. Je to stejné jako <strong
class="userinput"><code>m@(r,s)</code></strong>. Argumenty <code class="varname">r</code> a <code
class="varname">s</code> by měly být vektory se seznamy řádků a sloupců (nebo samostatná čísla, pokud požadu
jete jen jeden řádek nebo sloupec).</p></dd><dt><span class="term"><a
name="gel-function-SwapRows"></a>SwapRows</span></dt><dd><pre class="synopsis">SwapRows
(m,radek1,radek2)</pre><p>Prohodit dva řádky v matici.</p></dd><dt><span class="term"><a
name="gel-function-UpperTriangular"></a>UpperTriangular</span></dt><dd><pre class="synopsis">UpperTriangular
(M)</pre><p>Vrátit kopii matice <code class="varname">M</code> se všemi prvky pod diagonálou nastavenými na
nulu.</p></dd><dt><span class="term"><a name="gel-function-columns"></a>columns</span></dt><dd><pre
class="synopsis">columns (M)</pre><p>Vrátit počet sloupců matice.</p></dd><dt><span class="term"><a
name="gel-function-elements"></a>elements</span></dt><dd><pre class="synopsis">elements (M)</pre><p>Vrátit
celkový počet prvků matice. Tj. počet sloupců krát počet řádků.</p></dd><dt><span class="term"><a
name="gel-function-ones"></a>ones</span></dt><dd><pre class="synopsis">ones (radku,sloupcu...)</pr
e><p>Vytvořit matici ze samých jedniček (nebo řádkový vektor, pokud je zadán jen jeden argument). Když je
<code class="varname">radku</code> nebo <code class="varname">sloupcu</code> rovno nule, vrátí <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-rows"></a>rows</span></dt><dd><pre class="synopsis">rows (M)</pre><p>Vrátit počet řádků
matice.</p></dd><dt><span class="term"><a name="gel-function-zeros"></a>zeros</span></dt><dd><pre
class="synopsis">zeros (radku,sloupcu...)</pre><p>Vytvořit matici celou z nul (nebo řádkový vektor, pokud je
zadán jen jeden argument). Pokud je argument <code class="varname">radku</code> nebo <code
class="varname">sloupcu</code> roven nule, je vráceno <code
class="constant">null</code>.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
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header"><tr><th colspan="3" align="center">Lineární algebra</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s08.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
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iv><h2 class="title" style="clear: both"><a name="genius-gel-function-list-linear-algebra"></a>Lineární
algebra</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Získat pomocnou jednotkovou matici velikosti <code
class="varname">n</code>. Jde o čtvercovou matici ze samých nul vyjma diagonály, na které jsou jedničky. Je
to Jordanův blok s jedním vlastním číslem nula.</p><p>Více informací o Jordanově kanonické formě najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> (text je v angličtině) nebo
<a class="ulink" href="http://cs.wikipedia.org/wiki/Jordanova_norm%C3%A1ln%C3%AD_forma"; t
arget="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm
(v,A,w)</pre><p>Spočítat (v,w) vzhledem k bilineární formě dané maticí A.</p></dd><dt><span class="term"><a
name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Vrátit funkci takovou, že vyhodnocuje dva vektory vzhledem
k bilineární formě dané maticí A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Alternativní názvy: <code
class="function">CharPoly</code></p><p>Získat charakteristický polynom v podobě vektoru. Konkrétně vrací
koeficienty polynomu počínaje konstantním členem. Jedná se o polynom definovaný pomocí <strong
class="userinput"><code>det(M-xI)</code></strong>. Ko
řeny tohoto polynomu jsou vlastní čísla matice <code class="varname">M</code>. Viz <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">CharacteristicPolynomialFunction</a>.</p><p>Více
informací najdete v encyklopediích <a class="ulink"
href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> (text je v
anličtině) a <a class="ulink" href="http://planetmath.org/CharacteristicEquation";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Získat charakteristický polynom v podobě
funkce. Jedná se o polynom definovaný pomocí <strong class="userinput"><code>det(M-xI)</code></strong>.
Kořeny tohoto polynomu jsou vlastní čísla matice <code class="varname">M</code>. Viz <a class="link"
href="ch11s09
.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial";
target="_top">Wikipedia</a> (text je v anličtině) a <a class="ulink"
href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre class="synopsis">ColumnSpace
(M)</pre><p>Získat bázi matice pro prostor sloupců matice. Prakticky se vrátí matice, jejíž sloupce jsou
bázemi pro prostor sloupců matice <code class="varname">M</code>. To je prostor rozložený podle sloupců
matice <code class="varname">M</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="
gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre class="synopsis">CommutationMatrix
(m, n)</pre><p>Vrátit komutační matici <strong class="userinput"><code>K(m,n)</code></strong>, což je
jedinečná matice velikosti <strong class="userinput"><code>m*n</code></strong> krát <strong
class="userinput"><code>m*n</code></strong>, která splňuje <strong class="userinput"><code>K(m,n) *
MakeVector(A) = MakeVector(A.')</code></strong> pro všechny matice <code class="varname">A</code> velikosti
<code class="varname">m</code> krát <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre class="synopsis">CompanionMatrix
(p)</pre><p>Doplňková matice polynomu (jako vektor).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Konjugovaná transpozice matice (adju
ngovaná). Je to stejné jako operátor <strong class="userinput"><code>'</code></strong>.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="https://en.wikipedia.org/wiki/Conjugate_transpose";
target="_top">Wikipedia</a> (text je v angličtině) a <a class="ulink"
href="http://planetmath.org/ConjugateTranspose"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Alternativní názvy: <code class="function">convol</code></p><p>Spočítat konvoluci dvou
vodorovných vektorů.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Spočítat konvoluci dvou vodorovných vektorů. Výsledek vrátí
jako vektor a ne sečtené dohromady.</p></dd><dt><span class="term"><a name="gel-function-CrossProduct"></a>Cr
ossProduct</span></dt><dd><pre class="synopsis">CrossProduct (v,w)</pre><p>Vektorový součin dvou vektorů v
R<sup>3</sup> jako sloupcový vektor.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://cs.wikipedia.org/wiki/Vektorov%C3%BD_sou%C4%8Din";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre
class="synopsis">DeterminantalDivisorsInteger (M)</pre><p>Získat determinantové dělitele celočíselné
matice.</p></dd><dt><span class="term"><a name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre
class="synopsis">DirectSum (M,N...)</pre><p>Přímý součet matic.</p><p>Více informací najdete v encyklopedii
<a class="ulink"
href="https://cs.wikipedia.org/wiki/S%C4%8D%C3%ADt%C3%A1n%C3%AD_matic#Direktn.C3.AD_sou.C4.8Det";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function-DirectSumMatrixVector"></a
DirectSumMatrixVector</span></dt><dd><pre class="synopsis">DirectSumMatrixVector (v)</pre><p>Přímý součet
vektoru matic.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://cs.wikipedia.org/wiki/S%C4%8D%C3%ADt%C3%A1n%C3%AD_matic#Direktn.C3.AD_sou.C4.8Det";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Alternativní názvy: <code class="function">eig</code></p><p>Získat vlastní čísla čtvercové
matice. V současnosti pracuje pouze pro matice do velikosti 4 krát 4 nebo pro trojúhelníkové matice (pro
které jsou vlastní čísla na diagonále).</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a> (text je v
angličt
ině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Vlastn%C3%AD_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M,&vlastni_cisla)</pre><pre class="synopsis">Eigenvectors
(M, &vlastni_cisla, &nasobnosti)</pre><p>Získat vlastní vektory čtvercové matice. Volitelně získat
také vlastní čísla a jejich algebraické násobnosti. V současnosti pracuje pouze s maticemi do velikosti 2
krát 2.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Vlastn%C3%AD_%C4%8D%C3%ADslo";
target="_top">Wikipedia</
a>.</p></dd><dt><span class="term"><a name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre
class="synopsis">GramSchmidt (v,B...)</pre><p>Použít Gramův-Schmidtův proces (na sloupce) vzhledem k
unitárnímu prostoru danému <code class="varname">B</code>. Pokud <code class="varname">B</code> není zadáno,
je použit standardní hermitovský součin. <code class="varname">B</code> může být buď polybilineární funkce
dvou argumentů nebo to může být matice v polybilineární formě. Vektory budou vytvořeny ortogonální vzhledem k
<code class="varname">B</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/GramSchmidtOrthogonalization"; target="_top">Planetmath</a> (text je v angličtině)
a <a class="ulink" href="https://cs.wikipedia.org/wiki/Gramova-Schmidtova_ortogonalizace";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span></dt><dd
<pre class="synopsis">HankelMatrix (c,r)</pre><p>Henkelova matice, což je matice se stejnými vedlejšími
diagonálami. <code class="varname">c</code> je první řádek a <code class="varname">r</code> je poslední
sloupec. Předpokládá se, že oba argumenty budou vektory a poslední prvek <code class="varname">c</code>
bude stejný jako první prvek <code class="varname">r</code>.</p><p>Více informací najdete v encyklopedii <a
class="ulink" href="https://en.wikipedia.org/wiki/Hankel_matrix"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Hilbertova matice řádu <code class="varname">n</code>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix";
target="_top">Wikipedia</a> (text je v angličtině) a <a class="ulink" href="http://planetmath.org/Hilbert
Matrix" target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Image"></a>Image</span></dt><dd><pre class="synopsis">Image (T)</pre><p>Získat obraz
(sloupcový prostor) lineární transformace.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre
class="synopsis">InfNorm (v)</pre><p>Získat k vektoru normu typu nekonečno, někdy také nazývanou maximální
norma.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Získat invariantní činitele čtvercové celočíselné
matice.</p></dd><dt><span class="term"><a
name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatrix</span
</dt><dd><pre class="synopsis">InverseHilbertMatrix (n)</pre><p>Inverzní Hilbertova matice řádu <code
class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> (text je v angličtině) a <a
class="ulink" href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Je matice hermitovská? Tj. zda je rovna své konjugované transpozici.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://planetmath.org/HermitianMatrix";
target="_top">Planetmath</a> (text je v angličtině) a <a class="ulink"
href="https://en.wikipedia.org/wiki/Hermitian_matrix"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-IsInSubsp
ace"></a>IsInSubspace</span></dt><dd><pre class="synopsis">IsInSubspace (v,W)</pre><p>Zjistit, zda je vektor
v podprostoru.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Je matice (nebo číslo) invertovatelná (matice celých čísel je invertovatelná, když je
invertovatelná nad celými čísly)?</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Je matice (nebo číslo) invertovatelná nad
tělesem.</p></dd><dt><span class="term"><a name="gel-function-IsNormal"></a>IsNormal</span></dt><dd><pre
class="synopsis">IsNormal (M)</pre><p>Je <code class="varname">M</code> normální matice. To jest, zda <strong
class="userinput"><code>M*M' == M'*M</code></strong>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/NormalMatrix"; t
arget="_top">Planetmath</a> (text je v angličtině) nebo <a class="ulink"
href="http://mathworld.wolfram.com/NormalMatrix.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Je matice <code class="varname">M</code> hermitovská
pozitivně definitní matice? To znamená, zda je <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> vždy striktně pozitivní pro libovolný vektor
<code class="varname">v</code>. <code class="varname">M</code> musí být čtvercová a hermitovská, aby byla
pozitivně definitní. Kontrola, zda tomu tak je, spočívá v tom, zda každá hlavní podmatice má nezáporný
determinant. (Viz <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Poznamenejme, že někteří autoři
(např. Mathworld) nevyžadují, aby matice
<code class="varname">M</code> byla hermitovská a tak podmínka není skutečnu částí unitárního prostoru, ale
neberte to za dogma. Pokud chcete takovou kontrolu provést, jednoduše zkontrolujte hermitovskou část matice
<code class="varname">M</code> takto: <strong
class="userinput"><code>IsPositiveDefinite(M+M')</code></strong>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a>
(text je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Pozitivn%C4%9B_definitn%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Je matice <code class="varname">M</code> hermito
vská pozitivně semidefinitní matice? To znamená, zda je <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> vždy nezáporná pro libovolný vektor <code
class="varname">v</code>. <code class="varname">M</code> musí být čtvercová a hermitovská, aby byla pozitivně
semidefinitní. Kontrola, zda tomu tak je, spočívá v tom, zda každá hlavní podmatice má nezáporný determinant.
(Viz <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Poznamenejme, že někteří autoři
(např. Mathworld) nevyžadují, aby matice <code class="varname">M</code> byla hermitovská a tak podmínka není
skutečnu částí unitárního prostoru, ale neberte to za dogma. Pokud chcete takovou kontrolu provést, jednoduše
zkontrolujte hermitovskou část matice <code class="varname">M</code> takto: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Více informací najdete v
encyklopediích <
a class="ulink" href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> (text je v
angličtině) nebo <a class="ulink" href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html";
target="_top">Mathworld</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian
(M)</pre><p>Je matice antihermitovská? To znamená, zda je konjugovaná transpozice rovna negativní
matici.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/SkewHermitianMatrix"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre class="synopsis">IsUnitary (M)</pre><p>Je
matice unitární? To je, zda <strong class="userinput"><code>M'*M</code></strong> a <strong
class="userinput"><code>M*M'</code></strong> dají stejnou jednotkovou m
atici.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/UnitaryTransformation"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/UnitaryMatrix.html"; target="_top">Mathworld</a> (text je v
angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Unit%C3%A1rn%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-JordanBlock"></a>JordanBlock</span></dt><dd><pre class="synopsis">JordanBlock
(n,lambda)</pre><p>Alternativní názvy: <code class="function">J</code></p><p>Získat Jordanův blok
odpovídající vlastnímu číslu <code class="varname">lambda</code> s násobností <code
class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> (text je v angličtině),
<a class="ulink" href="http://mathworld.wolf
ram.com/JordanBlock.html" target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Jordanova_norm%C3%A1ln%C3%AD_forma";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre class="synopsis">Kernel (T)</pre><p>Získat jádro
(nulový prostor) lineární transformace.</p><p>(Viz <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Alternativní názvy: <code
class="function">TensorProduct</code></p><p>Spočítat Kroneckerův součin (tenzorový součin ve standardní bázi)
dvou matic.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://en.wikipedia.org/wiki/Kronecker_product"; target="_top">Wikipedia</a> (text je v angličtině), <a
class="ulink" href="http
://planetmath.org/KroneckerProduct" target="_top">Planetmath</a> (text je v angličtině) a <a class="ulink"
href="http://mathworld.wolfram.com/KroneckerProduct.html"; target="_top">Mathworld</a> (text je v
angličtině).</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>Získat LU rozklad matice <code class="varname">A</code> tak, že se najde dolní a horní
trojúhelníková matice, jejichž součinem je <code class="varname">A</code>. Výsledek se uloží v <code
class="varname">L</code> a <code class="varname">U</code>, což by měly být odkazy na proměnné. V případě
úspěchu vrací <code class="constant">true</code>. Například předpokládejme, že A je čtvercová matice, pak po
spuštění: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LUDecomposition(A,&L,&U)</
code></strong>
-</pre><p> budete mít dolní matici uloženou v proměnné s názvem <code class="varname">L</code> a horní matici
v proměnné s názvem <code class="varname">U</code>.</p><p>Jedná se o LU rozklad matice známý také jako
Croutův a/nebo Choleského rozklad. (ISBN 0-201-11577-8 pp.99-103) Horní trojúhelníková matice zahrnuje
diagonálu hodnot 1. Nejedná se o Doolittlovu metodu, která zahrnuje diagonálu jedniček do dolní
matice.</p><p>Ne všechny matice mají LU rozklad, například <strong
class="userinput"><code>[0,1;1,0]</code></strong> jej nemá a tato funkce v takovém případě vrátí <code
class="constant">false</code> a nastaví <code class="varname">L</code> a <code class="varname">U</code> na
<code class="constant">null</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/LUDecompo
sition.html" target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/LU_rozklad"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre class="synopsis">Minor
(M,i,j)</pre><p>Získat subdeterminant (též minor) <code class="varname">i</code>-<code
class="varname">j</code> matice.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/Minor"; target="_top">Planetmath</a>.</p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Vrátit sloupce matice, které nemají pivot.</p></dd><dt><span class="term"><a
name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm (v,p...)</pre><p>Alternativní
názvy: <code class="function">norm</code></p><p>Získat normu typu p (nebo typu 2, pokud není zadáno p)
vektoru.</p></
dd><dt><span class="term"><a name="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre
class="synopsis">NullSpace (T)</pre><p>Získat nulový prostor matice. Tj. jádro lineární transformace, která
matici představuje. Výsledek se vrací v podobě matice, jejíž sloupcový prostor je nulovým prostorem z <code
class="varname">T</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/Nullspace"; target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span
class="term"><a name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre class="synopsis">Nullity
(M)</pre><p>Alternativní názvy: <code class="function">nullity</code></p><p>Získat nulovost matice. Tzn.
vrátit rozměry nulového prostoru; rozměry jádra matice <code class="varname">M</code>.</p><p>Více informací
najdete v encyklopedii <a class="ulink" href="http://planetmath.org/Nullity"; target="_top">Planetmath</a>
(text je v angličtině).</p></dd><
dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre><p>Získat ortogonální doplněk sloupcového
prostoru.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Vrátit sloupce matice s pivoty, tzn. sloupce, které mají 1 v řádkově redukované podobě. Rovněž
vrací řádek, ve kterém se vyskytly.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Projekce vektoru <code class="varname">v</code> do podprostoru <code
class="varname">W</code> vzhledem k unitárnímu prostoru danému <code class="varname">B</code>. Pokud <code
class="varname">B</code> není zadáno, je použit standardní hermitovský součin. <code class="varname">B</code>
může být buď polybilineární funkce
dvou argumentů nebo to může být matice v polybilineární formě.</p></dd><dt><span class="term"><a
name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre class="synopsis">QRDecomposition
(A, Q)</pre><p>Získat QR rozklad čtvercové matice <code class="varname">A</code>, vrací horní trojúhelníkovou
matici <code class="varname">R</code> a nastavuje <code class="varname">Q</code> na ortogonální (unitární)
matici. <code class="varname">Q</code> by měl být odkaz na proměnnou nebo <code class="constant">null</code>,
pokud nic vrátit nechcete. Například pro </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>R = QRDecomposition(A,&Q)</code></strong>
-</pre><p> budete mít horní trojúhelníkovou matici uloženou v proměnné s názvem <code
class="varname">R</code> a ortogonální (unitární) matici v <code class="varname">Q</code>.</p><p>Více
informací najdete v encyklopediích <a class="ulink" href="http://planetmath.org/QRDecomposition";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/QRDecomposition.html"; target="_top">Mathworld</a> (text je v angličtině) a
<a class="ulink" href="https://cs.wikipedia.org/wiki/QR_rozklad";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Vrátit Rayleighův podíl (nazývaný také Rayleighův-Ritzův
koeficient nebo podíl) matice a vektoru.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> (text j
e v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)</pre><p>Najít vlastní čísla matice
<code class="varname">A</code> pomocí iterační metody Rayleighova podílu. <code class="varname">x</code> je
odhadovaný vlastní vektor a mohl by být náhodný. Měl by mít nenulovou imaginární část, pokud existuje nějaká
možnost, že budou nalezena komplexní vlastní čísla. Kód bude nanejvýše v <code class="varname">maxiter</code>
iteracích a vracet <code class="constant">null</code>, pokud není možné získat výsledek v rámci chyby <code
class="varname">epsilon</code>. <code class="varname">vecref</code> by měl být buď <code
class="constant">null</code> nebo odkaz na proměnnou, do které by se měl uložit vlastní vektor.</p><p>Více
informací o Rayleighově podíle najdete v encyklopedii
<a class="ulink" href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span></dt><dd><pre
class="synopsis">Rank (M)</pre><p>Alternativní názvy: <code class="function">rank</code></p><p>Získat hodnost
matice.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Vrátit Rosserovu matici, která je klasickým symetrickým problémem testu vlastního
čísla.</p></dd><dt><span class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre
class="synopsis">Rotation2D (úhel)</pre><p>Alternativní názvy: <code
class="function">RotationMatrix</code></p><p>Vrátit matici odpovídající otočení
okolo počátku v R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(úhel)</pre><p>Vrátit matici odpovídající otočení okolo počátku v R<sup>3</sup> kolem osy
x.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (úhel)</pre><p>Vrátit matici odpovídající otočení okolo počátku v R<sup>3</sup>
kolem osy y.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(úhel)</pre><p>Vrátit matici odpovídající otočení okolo počátku v R<sup>3</sup> kolem osy
z.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Získat bázi matice pro prostor řádků matice.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></
a>SesquilinearForm</span></dt><dd><pre class="synopsis">SesquilinearForm (v,A,w)</pre><p>Vyhodnotit (v,w)
vzhledem k polybilineární formě dané maticí <code class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Vrátit funkci vyhodnocující dva vektory vzhledem k
polybilineární formě dané maticí <code class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Vrátit Smithův kanonický tvar (normální forma) matice nad
poli (bude končit s jedničkami na diagonále).</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://en.wikipedia.org/wiki/Smith_normal_form"; target="_top">Wikipedia</a> (článek je v
angličtině).</p></dd><dt><span class="term"><a name=
"gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Vrátit Smithův kanonický tvar (normální formu) pro
čtvercové celočíselné matice nad celými čísly.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://en.wikipedia.org/wiki/Smith_normal_form"; target="_top">Wikipedia</a> (článek je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,argumenty...)</pre><p>Vyřešit lineární systém Mx=V, vrátit řešení V,
pokud existuje jedinečné řešení, jinak vrátit <code class="constant">null</code>. Je možné použít dva
dodatečné parametry předávané odkazem, ve kterých získáte redukované M a V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatr
ix (s, r...)</pre><p>Vrátit Teplitzovu matici sestavenou podle zadaného prvního sloupce <code
class="varname">c</code> a (volitelně) prvního řádku <code class="varname">r</code>. Pokud je zadán pouze
sloupec <code class="varname">c</code>, je pro první řádek použita konjugovaná a nekonjugovaná verze, aby se
získala hermitovská matice (samozřejmě za předpokladu, že je první prvek reálný).</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix";
target="_top">Wikipedia</a> (text je v angličtině) a <a class="ulink"
href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Alternativní názvy: <code class="function">trace</code></p><p>Spočítat
stopu matice. Jedná se o součet prvků na hlavní diagonále čtvercové matice.</p>
<p>Více informací najdete v encyklopediích <a class="ulink"
href="http://cs.wikipedia.org/wiki/Stopa_%28algebra%29"; target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/Trace"; target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span
class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre class="synopsis">Transpose
(M)</pre><p>Transponovat matici. Funkčně je to stejné, jako operátor <strong
class="userinput"><code>.'</code></strong>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://cs.wikipedia.org/wiki/Transpozice_matice"; target="_top">Wikipedia</a> a <a class="ulink"
href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span
class="term"><a name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alternativní názvy: <code class="function">vander</code></
p><p>Vrátit Vandermondovu matici.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Vandermondova_matice"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre
class="synopsis">VectorAngle (v,w,B...)</pre><p>Úhel dvou vektorů vzhledem k unitárnímu prostoru daného <code
class="varname">B</code>. Pokud <code class="varname">B</code> není zadáno, je použit standardní hermitovský
součin. <code class="varname">B</code> může být buď polybilineární funkce dvou argumentů nebo to může být
matice v polybilineární formě.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Přímý součet vektorových prostorů M a
N.</p></dd><dt><span class="term"><a name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceInter
section</span></dt><dd><pre class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Průnik podprostorů
daných pomocí M a N</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>Součet vektorových prostorů M a N, tj. {w | w=m+n, m in M, n
in N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Alternativní názvy: <code class="function">Adjugate</code></p><p>Získat
adjungovanou (reciproku) matici.</p></dd><dt><span class="term"><a
name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Alternativní názvy:
<code class="function">CREF</code> <code class="function">ColumnReducedEchelonForm</code></p><p>Spočítat
sloupcově odstupňovaný tvar matice.</p></dd><dt><span class="term"><a
name="gel-function-det"></a>det</span></dt><dd><pre class="synopsis">det (M
)</pre><p>Alternativní názvy: <code class="function">Determinant</code></p><p>Získat determinant
matice.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre class="synopsis">ref
(M)</pre><p>Alternativní názvy: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Získat řádkově odstupňovaný tvar matice. To jest, použít
Gaussovu eliminaci, ale bez zpětného dosazování do <code class="varname">M</code>. Nenulové řádky jsou
poděleny, aby všechny pivoty byly 1.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a> (text je v angličtině) a <a
class=
"ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alternativní názvy: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Získat redukovaný řádkově odstupňovaný tvar matice. To
jest, použít Gaussovu eliminaci se zpětným dosazováním do <code class="varname">M</code>.</p><p>Více
informací najdete v encyklopediích <a class="ulink"
href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form"; target="_top">Wikipedia</a> (text je v
angličtině) a <a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm";
target="_top">Planetmath</a> (text je v angličtině).</p></dd></dl></div></div><div
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header"><tr><th colspan="3" align="center">Lineární algebra</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s08.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
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iv><h2 class="title" style="clear: both"><a name="genius-gel-function-list-linear-algebra"></a>Lineární
algebra</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Získat pomocnou jednotkovou matici velikosti <code
class="varname">n</code>. Jde o čtvercovou matici ze samých nul vyjma diagonály, na které jsou jedničky. Je
to Jordanův blok s jedním vlastním číslem nula.</p><p>Více informací o Jordanově kanonické formě najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem";
target="_top">Planetmath</a> (text je v angličtině), <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> (text je v angličtině) nebo
<a class="ulink" href="http://cs.wikipedia.org/wiki/Jordanova_norm%C3%A1ln%C3%AD_forma"; t
arget="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm
(v,A,w)</pre><p>Spočítat (v,w) vzhledem k bilineární formě dané maticí A.</p></dd><dt><span class="term"><a
name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Vrátit funkci takovou, že vyhodnocuje dva vektory vzhledem
k bilineární formě dané maticí A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Alternativní názvy: <code
class="function">CharPoly</code></p><p>Získat charakteristický polynom v podobě vektoru. Konkrétně vrací
koeficienty polynomu počínaje konstantním členem. Jedná se o polynom definovaný pomocí <strong
class="userinput"><code>det(M-xI)</code></strong>. Ko
řeny tohoto polynomu jsou vlastní čísla matice <code class="varname">M</code>. Viz <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">CharacteristicPolynomialFunction</a>.</p><p>Více
informací najdete v encyklopediích <a class="ulink"
href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> (text je v
anličtině) a <a class="ulink" href="http://planetmath.org/CharacteristicEquation";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Získat charakteristický polynom v podobě
funkce. Jedná se o polynom definovaný pomocí <strong class="userinput"><code>det(M-xI)</code></strong>.
Kořeny tohoto polynomu jsou vlastní čísla matice <code class="varname">M</code>. Viz <a class="link"
href="ch11s09
.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial";
target="_top">Wikipedia</a> (text je v anličtině) a <a class="ulink"
href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre class="synopsis">ColumnSpace
(M)</pre><p>Získat bázi matice pro prostor sloupců matice. Prakticky se vrátí matice, jejíž sloupce jsou
bázemi pro prostor sloupců matice <code class="varname">M</code>. To je prostor rozložený podle sloupců
matice <code class="varname">M</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="
gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre class="synopsis">CommutationMatrix
(m, n)</pre><p>Vrátit komutační matici <strong class="userinput"><code>K(m,n)</code></strong>, což je
jedinečná matice velikosti <strong class="userinput"><code>m*n</code></strong> krát <strong
class="userinput"><code>m*n</code></strong>, která splňuje <strong class="userinput"><code>K(m,n) *
MakeVector(A) = MakeVector(A.')</code></strong> pro všechny matice <code class="varname">A</code> velikosti
<code class="varname">m</code> krát <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre class="synopsis">CompanionMatrix
(p)</pre><p>Doplňková matice polynomu (jako vektor).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Konjugovaná transpozice matice (adju
ngovaná). Je to stejné jako operátor <strong class="userinput"><code>'</code></strong>.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="https://en.wikipedia.org/wiki/Conjugate_transpose";
target="_top">Wikipedia</a> (text je v angličtině) a <a class="ulink"
href="http://planetmath.org/ConjugateTranspose"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Alternativní názvy: <code class="function">convol</code></p><p>Spočítat konvoluci dvou
vodorovných vektorů.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Spočítat konvoluci dvou vodorovných vektorů. Výsledek vrátí
jako vektor a ne sečtené dohromady.</p></dd><dt><span class="term"><a name="gel-function-CrossProduct"></a>Cr
ossProduct</span></dt><dd><pre class="synopsis">CrossProduct (v,w)</pre><p>Vektorový součin dvou vektorů v
R<sup>3</sup> jako sloupcový vektor.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://cs.wikipedia.org/wiki/Vektorov%C3%BD_sou%C4%8Din";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre
class="synopsis">DeterminantalDivisorsInteger (M)</pre><p>Získat determinantové dělitele celočíselné
matice.</p></dd><dt><span class="term"><a name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre
class="synopsis">DirectSum (M,N...)</pre><p>Přímý součet matic.</p><p>Více informací najdete v encyklopedii
<a class="ulink"
href="https://cs.wikipedia.org/wiki/S%C4%8D%C3%ADt%C3%A1n%C3%AD_matic#Direktn.C3.AD_sou.C4.8Det";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function-DirectSumMatrixVector"></a
DirectSumMatrixVector</span></dt><dd><pre class="synopsis">DirectSumMatrixVector (v)</pre><p>Přímý součet
vektoru matic.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://cs.wikipedia.org/wiki/S%C4%8D%C3%ADt%C3%A1n%C3%AD_matic#Direktn.C3.AD_sou.C4.8Det";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Alternativní názvy: <code class="function">eig</code></p><p>Získat vlastní čísla čtvercové
matice. V současnosti pracuje pouze pro matice do velikosti 4 krát 4 nebo pro trojúhelníkové matice (pro
které jsou vlastní čísla na diagonále).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M,&vlastni_cisla)</pre><pre class="synopsis">Eigenvectors
(M, &vlastni_cisla, &nasobnosti)</pre><p>Získat vlastní vektory čtvercové matice. Volitelně získat
také vlastní čísla a jejich algebraické násobnosti. V současnosti pracuje pouze s maticemi do velikosti 2
krát 2.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Použít Gramův-Schmidtův proces (na sloupce) vzhledem k unitárnímu prostoru danému <code
class="varname">B</code>. Pokud <code class="varname">B</code> není zadáno, je použit standardní hermitovský
součin. <code class="varname">B</code> může být buď polybilineární funkce dvou argumentů nebo to může být
matice v polybilineární formě. Vektory budou vytvořeny ortogonální vzhledem k <code
class="varname">B</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/GramSchmidtOrthogonalization"; target="_top">Planetmath</a> (text je v angličtině)
a <a class="ulink" href="https://cs.wikipedia.org/wiki/Gramova-Schmidtova_ortogonalizace";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span><
/dt><dd><pre class="synopsis">HankelMatrix (c,r)</pre><p>Henkelova matice, což je matice se stejnými
vedlejšími diagonálami. <code class="varname">c</code> je první řádek a <code class="varname">r</code> je
poslední sloupec. Předpokládá se, že oba argumenty budou vektory a poslední prvek <code
class="varname">c</code> bude stejný jako první prvek <code class="varname">r</code>.</p><p>Více informací
najdete v encyklopedii <a class="ulink" href="https://en.wikipedia.org/wiki/Hankel_matrix";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Hilbertova matice řádu <code class="varname">n</code>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix";
target="_top">Wikipedia</a> (text je v angličtině) a <a class="ulink" href="http://planetmath.org/
HilbertMatrix" target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Image"></a>Image</span></dt><dd><pre class="synopsis">Image (T)</pre><p>Získat obraz
(sloupcový prostor) lineární transformace.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre
class="synopsis">InfNorm (v)</pre><p>Získat k vektoru normu typu nekonečno, někdy také nazývanou maximální
norma.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Získat invariantní činitele čtvercové celočíselné
matice.</p></dd><dt><span class="term"><a name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatri
x</span></dt><dd><pre class="synopsis">InverseHilbertMatrix (n)</pre><p>Inverzní Hilbertova matice řádu
<code class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> (text je v angličtině) a <a
class="ulink" href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Je matice hermitovská? Tj. zda je rovna své konjugované transpozici.</p><p>Více informací najdete
v encyklopediích <a class="ulink" href="http://planetmath.org/HermitianMatrix"; target="_top">Planetmath</a>
(text je v angličtině) a <a class="ulink" href="https://en.wikipedia.org/wiki/Hermitian_matrix";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a name="gel-function-Is
InSubspace"></a>IsInSubspace</span></dt><dd><pre class="synopsis">IsInSubspace (v,W)</pre><p>Zjistit, zda je
vektor v podprostoru.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Je matice (nebo číslo) invertovatelná (matice celých čísel je invertovatelná, když je
invertovatelná nad celými čísly)?</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Je matice (nebo číslo) invertovatelná nad
tělesem.</p></dd><dt><span class="term"><a name="gel-function-IsNormal"></a>IsNormal</span></dt><dd><pre
class="synopsis">IsNormal (M)</pre><p>Je <code class="varname">M</code> normální matice. To jest, zda <strong
class="userinput"><code>M*M' == M'*M</code></strong>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/NormalMa
trix" target="_top">Planetmath</a> (text je v angličtině) nebo <a class="ulink"
href="http://mathworld.wolfram.com/NormalMatrix.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Je matice <code class="varname">M</code> hermitovská
pozitivně definitní matice? To znamená, zda je <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> vždy striktně pozitivní pro libovolný vektor
<code class="varname">v</code>. <code class="varname">M</code> musí být čtvercová a hermitovská, aby byla
pozitivně definitní. Kontrola, zda tomu tak je, spočívá v tom, zda každá hlavní podmatice má nezáporný
determinant. (Viz <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Poznamenejme, že někteří autoři
(např. Mathworld) nevyžadují, aby
matice <code class="varname">M</code> byla hermitovská a tak podmínka není skutečnu částí unitárního
prostoru, ale neberte to za dogma. Pokud chcete takovou kontrolu provést, jednoduše zkontrolujte hermitovskou
část matice <code class="varname">M</code> takto: <strong
class="userinput"><code>IsPositiveDefinite(M+M')</code></strong>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a>
(text je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Pozitivn%C4%9B_definitn%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Je matice <code class="varname">M</code>
hermitovská pozitivně semidefinitní matice? To znamená, zda je <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> vždy nezáporná pro libovolný vektor <code
class="varname">v</code>. <code class="varname">M</code> musí být čtvercová a hermitovská, aby byla pozitivně
semidefinitní. Kontrola, zda tomu tak je, spočívá v tom, zda každá hlavní podmatice má nezáporný determinant.
(Viz <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Poznamenejme, že někteří autoři
(např. Mathworld) nevyžadují, aby matice <code class="varname">M</code> byla hermitovská a tak podmínka není
skutečnu částí unitárního prostoru, ale neberte to za dogma. Pokud chcete takovou kontrolu provést, jednoduše
zkontrolujte hermitovskou část matice <code class="varname">M</code> takto: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Více informací najdete v
encykloped
iích <a class="ulink" href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> (text
je v angličtině) nebo <a class="ulink" href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html";
target="_top">Mathworld</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian
(M)</pre><p>Je matice antihermitovská? To znamená, zda je konjugovaná transpozice rovna negativní
matici.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/SkewHermitianMatrix"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre class="synopsis">IsUnitary (M)</pre><p>Je
matice unitární? To je, zda <strong class="userinput"><code>M'*M</code></strong> a <strong
class="userinput"><code>M*M'</code></strong> dají stejnou jednot
kovou matici.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/UnitaryTransformation"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/UnitaryMatrix.html"; target="_top">Mathworld</a> (text je v
angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Unit%C3%A1rn%C3%AD_matice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-JordanBlock"></a>JordanBlock</span></dt><dd><pre class="synopsis">JordanBlock
(n,lambda)</pre><p>Alternativní názvy: <code class="function">J</code></p><p>Získat Jordanův blok
odpovídající vlastnímu číslu <code class="varname">lambda</code> s násobností <code
class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> (text je v angličtině),
<a class="ulink" href="http://mathwor
ld.wolfram.com/JordanBlock.html" target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Jordanova_norm%C3%A1ln%C3%AD_forma";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre class="synopsis">Kernel (T)</pre><p>Získat jádro
(nulový prostor) lineární transformace.</p><p>(Viz <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Alternativní názvy: <code
class="function">TensorProduct</code></p><p>Spočítat Kroneckerův součin (tenzorový součin ve standardní bázi)
dvou matic.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>Získat LU rozklad matice <code class="varname">A</code> tak, že se najde dolní a horní
trojúhelníková matice, jejichž součinem je <code class="varname">A</code>. Výsledek se uloží v <code
class="varname">L</code> a <code class="varname">U</code>, což by měly být odkazy na proměnné. V případě
úspěchu vrací <code class="constant">true</code>. Například předpokládejme, že A je čtvercová matice, pak po
spuštění: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LUDecomposition(A,&L,&U)</code></strong>
+</pre><p> budete mít dolní matici uloženou v proměnné s názvem <code class="varname">L</code> a horní matici
v proměnné s názvem <code class="varname">U</code>.</p><p>Jedná se o LU rozklad matice známý také jako
Croutův a/nebo Choleského rozklad. (ISBN 0-201-11577-8 pp.99-103) Horní trojúhelníková matice zahrnuje
diagonálu hodnot 1. Nejedná se o Doolittlovu metodu, která zahrnuje diagonálu jedniček do dolní
matice.</p><p>Ne všechny matice mají LU rozklad, například <strong
class="userinput"><code>[0,1;1,0]</code></strong> jej nemá a tato funkce v takovém případě vrátí <code
class="constant">false</code> a nastaví <code class="varname">L</code> a <code class="varname">U</code> na
<code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Získat subdeterminant (též minor) <code class="varname">i</code>-<code
class="varname">j</code> matice.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/Minor"; target="_top">Planetmath</a>.</p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Vrátit sloupce matice, které nemají pivot.</p></dd><dt><span class="term"><a
name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm (v,p...)</pre><p>Alternativní
názvy: <code class="function">norm</code></p><p>Získat normu typu p (nebo typu 2, pokud není zadáno p)
vektoru.</p></dd><dt><span class="term"><a name="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre
class="synopsis">NullSpace (T)</pre><p>Získat nulový prostor ma
tice. Tj. jádro lineární transformace, která matici představuje. Výsledek se vrací v podobě matice, jejíž
sloupcový prostor je nulovým prostorem z <code class="varname">T</code>.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="http://planetmath.org/Nullspace"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre
class="synopsis">Nullity (M)</pre><p>Alternativní názvy: <code class="function">nullity</code></p><p>Získat
nulovost matice. Tzn. vrátit rozměry nulového prostoru; rozměry jádra matice <code
class="varname">M</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/Nullity"; target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span
class="term"><a name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre>
<p>Získat ortogonální doplněk sloupcového prostoru.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Vrátit sloupce matice s pivoty, tzn. sloupce, které mají 1 v řádkově redukované podobě. Rovněž
vrací řádek, ve kterém se vyskytly.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Projekce vektoru <code class="varname">v</code> do podprostoru <code
class="varname">W</code> vzhledem k unitárnímu prostoru danému <code class="varname">B</code>. Pokud <code
class="varname">B</code> není zadáno, je použit standardní hermitovský součin. <code class="varname">B</code>
může být buď polybilineární funkce dvou argumentů nebo to může být matice v polybilineární
formě.</p></dd><dt><span class="term"><a name="gel-function-QRDecomposition"></a>QRDecompositio
n</span></dt><dd><pre class="synopsis">QRDecomposition (A, Q)</pre><p>Získat QR rozklad čtvercové matice
<code class="varname">A</code>, vrací horní trojúhelníkovou matici <code class="varname">R</code> a nastavuje
<code class="varname">Q</code> na ortogonální (unitární) matici. <code class="varname">Q</code> by měl být
odkaz na proměnnou nebo <code class="constant">null</code>, pokud nic vrátit nechcete. Například pro </p><pre
class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>R =
QRDecomposition(A,&Q)</code></strong>
+</pre><p> budete mít horní trojúhelníkovou matici uloženou v proměnné s názvem <code
class="varname">R</code> a ortogonální (unitární) matici v <code class="varname">Q</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Vrátit Rayleighův podíl (nazývaný také Rayleighův-Ritzův
koeficient nebo podíl) matice a vektoru.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)</pre><p>Najít vlastní čísla matice
<code class="varname">A</code> pomocí iterační metody Rayleighova podílu. <code class="varname">x</code> je
odhadovaný vlastní vektor a mohl by být náhodný. Měl by mít nenulovou imaginární část, pokud existuje nějaká
možnost, že budou nalezena komplexní vlastní čísla. Kód bude nanejvýše
v <code class="varname">maxiter</code> iteracích a vracet <code class="constant">null</code>, pokud není
možné získat výsledek v rámci chyby <code class="varname">epsilon</code>. <code class="varname">vecref</code>
by měl být buď <code class="constant">null</code> nebo odkaz na proměnnou, do které by se měl uložit vlastní
vektor.</p><p>Více informací o Rayleighově podíle najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span></dt><dd><pre
class="synopsis">Rank (M)</pre><p>Alternativní názvy: <code class="function">rank</code></p><p>Získat hodnost
matice.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-RosserMatrix"></a>RosserMatrix<
/span></dt><dd><pre class="synopsis">RosserMatrix ()</pre><p>Vrátit Rosserovu matici, která je klasickým
symetrickým problémem testu vlastního čísla.</p></dd><dt><span class="term"><a
name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(úhel)</pre><p>Alternativní názvy: <code class="function">RotationMatrix</code></p><p>Vrátit matici
odpovídající otočení okolo počátku v R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(úhel)</pre><p>Vrátit matici odpovídající otočení okolo počátku v R<sup>3</sup> kolem osy
x.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (úhel)</pre><p>Vrátit matici odpovídající otočení okolo počátku v R<sup>3</sup>
kolem osy y.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DZ"></a>Rota
tion3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ (úhel)</pre><p>Vrátit matici odpovídající otočení
okolo počátku v R<sup>3</sup> kolem osy z.</p></dd><dt><span class="term"><a
name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre class="synopsis">RowSpace (M)</pre><p>Získat
bázi matice pro prostor řádků matice.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Vyhodnotit (v,w) vzhledem k polybilineární formě dané
maticí <code class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Vrátit funkci vyhodnocující dva vektory vzhledem k
polybilineární formě dané maticí <code class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>S
mithNormalFormField</span></dt><dd><pre class="synopsis">SmithNormalFormField (A)</pre><p>Vrátit Smithův
kanonický tvar (normální forma) matice nad poli (bude končit s jedničkami na diagonále).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Vrátit Smithův kanonický tvar (normální formu) pro
čtvercové celočíselné matice nad celými čísly.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,argumenty...)</pre><p>Vyřešit lineární systém Mx=V, vrátit řešení V,
pokud existuje jedinečné řešení, jinak vrátit <code class="constant">null</code>. Je možné použít dva
dodatečné parametry předávané odkazem, ve kterých získáte redukované M a V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (s,
r...)</pre><p>Vrátit Teplitzovu matici sestavenou podle zadaného prvního sloupce <code
class="varname">c</code> a (volitelně) prvního řádku <code class="varname">r</code>. Pokud je zadán pouze
sloupec <code class="varname">c</code>, je pro první řádek použita konjugovaná a nekonjugovaná verze, aby se
získala hermitovská matice (samozřejmě za předpokladu, že je první prvek reálný)
.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Alternativní názvy: <code class="function">trace</code></p><p>Spočítat
stopu matice. Jedná se o součet prvků na hlavní diagonále čtvercové matice.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Transponovat matici. Funkčně je to stejné, jako operátor <strong
class="userinput"><code>.'</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alternativní názvy: <code
class="function">vander</code></p><p>Vrátit Vandermondovu matici.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>Úhel dvou vektorů vzhledem k unitárnímu prostoru daného <code class="varname">B</code>.
Pokud <code class="varname">B</code> není zadáno, je použit standardní hermitovský součin. <code
class="varname">B</code> může být buď polybilineární funkce dvou argumentů nebo to může být matice v
polybilineární formě.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Přímý součet vektorových prostorů M a
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Průnik podprostorů daných pomocí M a
N</p></dd><dt><span class="term"><a name="gel-
function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre class="synopsis">VectorSubspaceSum
(M,N)</pre><p>Součet vektorových prostorů M a N, tj. {w | w=m+n, m in M, n in N}.</p></dd><dt><span
class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre class="synopsis">adj
(m)</pre><p>Alternativní názvy: <code class="function">Adjugate</code></p><p>Získat adjungovanou (reciproku)
matici.</p></dd><dt><span class="term"><a name="gel-function-cref"></a>cref</span></dt><dd><pre
class="synopsis">cref (M)</pre><p>Alternativní názvy: <code class="function">CREF</code> <code
class="function">ColumnReducedEchelonForm</code></p><p>Spočítat sloupcově odstupňovaný tvar
matice.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Alternativní názvy: <code class="function">Determinant</code></p><p>Získat
determinant matice.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Alternativní názvy: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Získat řádkově odstupňovaný tvar matice. To jest, použít
Gaussovu eliminaci, ale bez zpětného dosazování do <code class="varname">M</code>. Nenulové řádky jsou
poděleny, aby všechny pivoty byly 1.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alternativní názvy: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Získat redukovaný řádkově odstupňovaný tvar matice. To
jest, použít Gaussovu eliminaci se zpětným dosazováním do <code class="varname">M</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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name="genius-gel-function-list-combinatorics"></a>Kombinatorika</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Získat <code
class="varname">n</code>-té Catalanovo číslo.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Catalanova_%C4%8D%C3%ADsla";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Získat jako vektor vektorů všechny kombinace k-té třídy z prvků 1 až n. (Viz také <a
class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p></dd><
dt><span class="term"><a name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre
class="synopsis">DoubleFactorial (n)</pre><p>Dvojitý faktoriál: <strong
class="userinput"><code>n(n-2)(n-4)…</code></strong></p><p>Více informací najdete v encyklopedii <a
class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre class="synopsis">Factorial
(n)</pre><p>Faktoriál: <strong class="userinput"><code>n(n-1)(n-2)…</code></strong></p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a>
(text je v angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Faktori%C3%A1l";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre class="synopsi
s">FallingFactorial (n,k)</pre><p>Klesající faktoriál: <strong class="userinput"><code>(n)_k =
n(n-1)…(n-(k-1))</code></strong></p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre class="synopsis">Fibonacci
(x)</pre><p>Alternativní názvy: <code class="function">fib</code></p><p>Vypočítat <code
class="varname">n</code>-té Fibonacciho číslo. Tj. číslo definované rekurzivně jako <strong
class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</code></strong> a <strong
class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a>
(text je v angličtině), <a class="ulink" href="http://mathworl
d.wolfram.com/FibonacciNumber.html" target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Fibonacciho_posloupnost"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre
class="synopsis">FrobeniusNumber (v,arg...)</pre><p>Spočítat Frobeniusovo číslo. Tzn. spočítat nejmenší
číslo, které nemůže být dáno jako lineární kombinace celých nezáporných čísel zadaných jako vektor
nezáporných celých čísel. Vektor může být zadán jako samostatná čísla nebo jeden vektor. Všechna zadaná čísla
by měla mít největšího společného dělitele 1.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://mathworld.wolfram.com/FrobeniusNumber.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pr
e class="synopsis">GaloisMatrix (kombinacni_pravidlo)</pre><p>Galoisova matice daná lineárním kombinačním
pravidlem (a_1*x_1+…+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Najít takový vektor <code class="varname">c</code> nezáporných celých čísel, že skalární součin
s <code class="varname">v</code> je roven <code class="varname">n</code>. Když to není možné, vrátí <code
class="constant">null</code>. Vektor <code class="varname">v</code> by měl být předán seřazený ve vzestupném
pořadí a měl by se skládat z nezáporných celých čísel.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html"; target="_top">Mathworld</a> (text je v
angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Hladov%C3%BD_algoritmus";
target="_top">Wikipedia
</a>.</p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alternativní názvy: <code class="function">HarmonicH</code></p><p>Harmonické číslo, <code
class="varname">n</code>-té harmonické číslo řádu <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadterova funkce q(n) definovaná jako q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2))</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (pocatecni_hodnoty,kombinacni_pravidlo,n)</pre><p>Spočítat lineární
rekurzivní posloupnost pomocí Galoisova krokování.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multi
nomial (v,arg...)</pre><p>Spočítat multinomické koeficienty. Přebírá vektor <code class="varname">k</code>
nezáporných celých čísel a spočítá multinomický koeficient. To odpovídá koeficientu v homogenním polynomu v
<code class="varname">k</code> proměnných s odpovídajícími mocninami.</p><p>Vzorec pro <strong
class="userinput"><code>Multinomial(a,b,c)</code></strong> se dá napsat jako: </p><pre
class="programlisting">(a+b+c)! / (a!b!c!)
-</pre><p> Jinými slovy, pokud máme jen dva prvky, pak <strong
class="userinput"><code>Multinomial(a,b)</code></strong> je to stejné, jako <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> nebo <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a> (text je v
angličtině), <a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> (text je v angličtině) a <a class="ulink"
href="https://cs.wikipedia.org/wiki/Multinomick%C3%A1_v%C4%9Bta";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Získat kombinaci, která by následovala po kombinaci <code class="varname">v</code> v pořadí
kombinací, první kombinací by měla být
<strong class="userinput"><code>[1:k]</code></strong>. To je užitečné, pokud máte hodně kombinací, které
chcete projít a nechcete plýtvat pamětí na uložení všech.</p><p>S funkcí Combinations byste normálně napsali
smyčku jako: </p><pre class="screen"><strong class="userinput"><code>for n in Combinations (4,6) do (
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Kombinatorika</title><meta name="generator" content="DocBook XSL Stylesheets
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width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Kombinatorika</th></tr><tr><td
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align="center">Kapitola 11. Seznam funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s11.html">Další</a></td></tr></table><hr></div><div class="sect1"><div class="title
page"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-combinatorics"></a>Kombinatorika</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Získat <code
class="varname">n</code>-té Catalanovo číslo.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Catalanova_%C4%8D%C3%ADsla";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Získat jako vektor vektorů všechny kombinace k-té třídy z prvků 1 až n. (Viz také <a
class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Dvojitý faktoriál: <strong class="userinput"><code>n(n-2)(n-4)…</code></strong></p><p>Více
informací najdete v encyklopedii <a class="ulink" href="http://planetmath.org/DoubleFactorial";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre class="synopsis">Factorial
(n)</pre><p>Faktoriál: <strong class="userinput"><code>n(n-1)(n-2)…</code></strong></p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a>
(text je v angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Faktori%C3%A1l";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre c
lass="synopsis">FallingFactorial (n,k)</pre><p>Klesající faktoriál: <strong class="userinput"><code>(n)_k =
n(n-1)…(n-(k-1))</code></strong></p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre class="synopsis">Fibonacci
(x)</pre><p>Alternativní názvy: <code class="function">fib</code></p><p>Vypočítat <code
class="varname">n</code>-té Fibonacciho číslo. Tj. číslo definované rekurzivně jako <strong
class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</code></strong> a <strong
class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
+ Calculate the Frobenius number. That is calculate largest
+ number that cannot be given as a non-negative integer linear
+ combination of a given vector of non-negative integers.
+ The vector can be given as separate numbers or a single vector.
+ All the numbers given should have GCD of 1.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(kombinacni_pravidlo)</pre><p>Galoisova matice daná lineárním kombinačním pravidlem
(a_1*x_1+…+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Najít takový vektor <code class="varname">c</code> nezáporných celých čísel, že skalární součin
s <code class="varname">v</code> je roven <code class="varname">n</code>. Když to není možné, vrátí <code
class="constant">null</code>. Vektor <code class="varname">v</code> by měl být předán seřazený ve vzestupném
pořadí a měl by se skládat z nezáporných celých čísel.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alternativní názvy: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadterova funkce q(n) definovaná jako q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2))</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (pocatecni_hodnoty,kombinacni_pravidlo,n)</pre><p>Spočítat lineární
rekurzivní posloupnost pomocí Galoisova krokování.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Spočítat multinomické koeficienty. Přebírá vektor <code class="varname">k</code>
nezáporných celých čísel a spočítá multinomický koeficient. To odpovídá koeficientu v homogenním polynomu v
<code class="varname">k</code> proměnných s odpovídajícími mocninami.</p><p>Vzorec pro <strong
class="userinput"><code>Multinomial(a,b,c)</code></strong> se dá napsat jako: </p><pre
class="programlisting">(a+b+c)! / (a!b!c!)
+</pre><p> Jinými slovy, pokud máme jen dva prvky, pak <strong
class="userinput"><code>Multinomial(a,b)</code></strong> je to stejné, jako <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> nebo <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Získat kombinaci, která by následovala po kombinaci <code class="varname">v</code> v pořadí
kombinací, první kombinací by měla být <strong class="userinput"><code>[1:k]</code></strong>. To je užitečné,
pokud máte hodně kombinací, které chcete projít a nechcete plýtvat pamětí na uložení všech.</p><p>S funkcí
Combinations byste normálně napsali smyčku jako: </p><pre class="screen"><strong class="userinput"><code>for
n in Combinations (4,6) do (
NejakaFunkce (n)
);</code></strong>
</pre><p> Ale s funkcí NextCombination byste napsali něco takového: </p><pre class="screen"><strong
class="userinput"><code>n:=[1:4];
do (
NejakaFunkce (n)
) while not IsNull(n:=NextCombination(n,6));</code></strong>
-</pre><p> Viz <a class="link"
href="ch11s10.html#gel-function-Combinations">Combinations</a>.</p></dd><dt><span class="term"><a
name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre class="synopsis">Pascal (i)</pre><p>Získat Pascalův
trojúhelník v podobě matice. Vrátí dolní trojúhelníkovou matici <code class="varname">i</code>+1 krát <code
class="varname">i</code>+1, která je Pascalovým trojúhelníkem po <code class="varname">i</code>
iteracích.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Pascal%C5%AFv_troj%C3%BAheln%C3%ADk";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Získat jako vektor vektorů všechny variace <code class="varname">k</code>-té tříd
y z prvků 1 až <code class="varname">n</code> prvků, případně permutace pro <code
class="varname">k</code>=<code class="varname">n</code>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> (text je v
angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Permutace"; target="_top">Wikipedia</a>
(permutace) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Variace_%28kombinatorika%29";
target="_top">Wikipedia</a> (variace).</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alternativní názvy: <code class="function">Pochhammer</code></p><p>(Pochhammerův) stoupacící
faktoriál: (n)_k = n(n+1)…(n+(k-1))</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alternativní názvy: <code
class="function">StirlingS1</code></p><p>Stirlingovo číslo prvního druhu.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/StirlingNumbersOfTheFirstKind";
target="_top">Planetmath</a> (text je v angličtině) nebo <a class="ulink"
href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Alternativní názvy: <code
class="function">StirlingS2</code></p><p>Stirlingovo číslo druhého druhu.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/Stir
lingNumbersSecondKind" target="_top">Planetmath</a> (text je v angličtině) nebo <a class="ulink"
href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Subfaktoriál: n! krát suma_{k=0}^n (-1)^k/k!</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangular
(n)</pre><p>Spočítat <code class="varname">n</code>-té trojúhelníkové číslo.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/TriangularNumbers"; target="_top">Planetmath</a>
(text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADkov%C3%A9_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function-nCr"></a>
nCr</span></dt><dd><pre class="synopsis">nCr (n,r)</pre><p>Alternativní názvy: <code
class="function">Binomial</code></p><p>Spočítat kombinace, tj. kombinační číslo. <code
class="varname">n</code> může být libovolné reálné číslo.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Kombina%C4%8Dn%C3%AD_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-nPr"></a>nPr</span></dt><dd><pre class="synopsis">nPr (n,k)</pre><p>Spočítat počet variací
<code class="varname">k</code>-té třídy z prvků 1 až <code class="varname">n</code>, respektive počet
permutací při <code class="varname">k</code> rovno <code class="varname">n</code>.</p><p>Více informací
najdete v encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target
="_top">Mathworld</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Permutace"; target="_top">Wikipedia</a> (permutace) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Variace_%28kombinatorika%29"; target="_top">Wikipedia</a>
(variace).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Lineární
algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
align="right" valign="top"> Diferenciální/integrální počet </td></tr></table></div></body></html>
+</pre><p> Viz <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Získat Pascalův trojúhelník v podobě matice. Vrátí dolní trojúhelníkovou
matici <code class="varname">i</code>+1 krát <code class="varname">i</code>+1, která je Pascalovým
trojúhelníkem po <code class="varname">i</code> iteracích.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Pascal%C5%AFv_troj%C3%BAheln%C3%ADk";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Získat jako vektor vektorů všechny variace <code class="varname">k</code>-té třídy z prvků 1 až
<code class="varname">n</code> prvků, případně permutace pro <code cl
ass="varname">k</code>=<code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alternativní názvy: <code class="function">Pochhammer</code></p><p>(Pochhammerův) stoupacící
faktoriál: (n)_k = n(n+1)…(n+(k-1))</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alternativní názvy: <code
class="function">StirlingS1</code></p><p>Stirlingovo číslo prvního druhu.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/StirlingNumbersOfTheFirstKind";
target="_top">Planetmath</a> (text je v angličtině) nebo <a class="ulink"
href="http://mathworld.wolfram.com/StirlingNumberofth
eFirstKind.html" target="_top">Mathworld</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Alternativní názvy: <code
class="function">StirlingS2</code></p><p>Stirlingovo číslo druhého druhu.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/StirlingNumbersSecondKind";
target="_top">Planetmath</a> (text je v angličtině) nebo <a class="ulink"
href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html"; target="_top">Mathworld</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Subfaktoriál: n! krát suma_{k=0}^n (-1)^k/k!</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangula
r (n)</pre><p>Spočítat <code class="varname">n</code>-té trojúhelníkové číslo.</p><p>Více informací najdete
v encyklopediích <a class="ulink" href="http://planetmath.org/TriangularNumbers"; target="_top">Planetmath</a>
(text je v angličtině) a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADkov%C3%A9_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-nCr"></a>nCr</span></dt><dd><pre class="synopsis">nCr (n,r)</pre><p>Alternativní názvy:
<code class="function">Binomial</code></p><p>Spočítat kombinace, tj. kombinační číslo. <code
class="varname">n</code> může být libovolné reálné číslo.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Kombina%C4%8Dn%C3%AD_%C4%8D%C3%ADslo";
target="_top">Wikipedia</a>.</p></dd><dt><span cl
ass="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre class="synopsis">nPr
(n,k)</pre><p>Spočítat počet variací <code class="varname">k</code>-té třídy z prvků 1 až <code
class="varname">n</code>, respektive počet permutací při <code class="varname">k</code> rovno <code
class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Lineární
algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
align="right" valign="top"> Diferenciální/integrální počet </td></tr></table></div></body></html>
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charset=UTF-8"><title>Diferenciální/integrální počet</title><meta name="generator" content="DocBook XSL
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</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch11s10.html">Předcházející</a> </td><th
width="60%" align="center">Kapitola 11. Seznam funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s12.html">Další</a></td></tr></table><hr></div><div class="sect1"><div c
lass="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-calculus"></a>Diferenciální/integrální počet </h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integrovat f složeným Simpsonovým pravidlem na
intervalu [a,b] s n podintervaly s chybou podle max(f'''')*h^4*(b-a)/180. Upozorňujeme, že n by mělo být
sudé.</p><p>Více informací najdete v encyklopedii <a class="ulink" href="http://planetmath.org/SimpsonsRule";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance (f,a,b,omezeni_ctvrte_derivace,tolerance)</pre><p>Integrovat
f složeným Simpson
ovým pravidlem na intervalu [a,b] s počtem kroků počítaným podle omezení čtvrté derivace a podle požadované
tolerance.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Zkusit spočítat derivaci, nejprve symbolicky a pak numericky.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="https://cs.wikipedia.org/wiki/Derivace";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Vrátit funkci, která je sudým periodickým rozšířením
<code class="function">f</code> s poloviční periodou <code class="varname">L</code>. Tj. funkce definovaná na
inter
valu <strong class="userinput"><code>[0,L]</code></strong> rozšířená, aby byla sudá na <strong
class="userinput"><code>[-L,L]</code></strong> a pak rozšířená, aby byla periodická s periodou <strong
class="userinput"><code>2*L</code></strong>.</p><p>Viz také <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> a <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Vrátit funkci, která je Fourierovu řadou s koeficienty
danými vektory <code class="varname">a</code> (sinové) a <code class="varname">b</code> (kosinové). Vezměte
na vědomí, že <strong class="userinput"><code>a@(1)</code></strong> je konstantní koeficient! To znamená, že
<strong class="userinput"><code>a@(n)</
code></strong> odkazuje na člen <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, zatímco
<strong class="userinput"><code>b@(n)</code></strong> odkazuje na člen <strong
class="userinput"><code>sin(x*n*pi/L)</code></strong>. Buďto <code class="varname">a</code> nebo <code
class="varname">b</code> může být <code class="constant">null</code>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre
class="synopsis">InfiniteProduct (fce,start,prirustek)</pre><p>Zkusit spočítat nekonečný součin funkce s
jedním parametrem.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></d
t><dd><pre class="synopsis">InfiniteProduct2 (fce,arg,start,prirustek)</pre><p>Zkusit spočítat nekonečný
součin funkce se dvěma parametry s fce (arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(fce,start,prirustek)</pre><p>Zkusit spočítat nekonečný součet funkce s jedním parametrem.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (fce,arg,start,prirustek)</pre><p>Zkusit spočítat nekonečný součet funkce se
dvěma parametry s fce (arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Zkusit zjistit pomocí výpočtu limity v x0, jestli je funkce reálné proměnné v tomto bodě
spojitá.</p></dd><dt><span class="term"><a name="gel-function-IsDifferentiable"></a>IsDifferentiable</span>
</dt><dd><pre class="synopsis">IsDifferentiable (f,x0)</pre><p>Otestovat na diferencovatelnost aproximací
limit zleva a zprava a porovnáním.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Spočítat limitu zleva funkce reálné proměnné v x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Spočítat limitu
funkce reálné proměnné v x0. Zkusí vypočítat limitu zleva i zprava.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Integrovat trojúhelníkovou metodou (pravidlem prostředního bodu).</p></dd><dt><span
class="term"><a name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alternativní názvy: <code class="
function">NDerivative</code></p><p>Zkusit vypočítat numerickou derivaci.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="https://cs.wikipedia.org/wiki/Derivace";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Vrátit vektor vektorů <strong
class="userinput"><code>[a,b]</code></strong>, kde <code class="varname">a</code> jsou kosinové koeficienty a
<code class="varname">b</code> sinové koeficienty Fourierovy řady funkce <code class="function">f</code> s
poloviční periodou <code class="varname">L</code> (tj. definovanou na <strong
class="userinput"><code>[-L,L]</code></strong> a periodicky rozšířenou) s numericky spočítanými koeficienty
do <code class="varname">N</code>-té harmonické. Koeficienty jsou spočítány numerickou integrací pomocí <a
class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> (text je v angličtině)
nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Vrátit funkci, která je Fourierovou řadou
funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code> (tj. definovanou
na <strong class="userinput"><code>[-L,L]</code></strong> a periodicky rozšířenou) s numericky spočítanými
koeficienty do <code class="varname">N</code>-té harmonické. Jde o čistě trigonom
etrickou řadu složenou ze sinů a kosinů. Koeficienty jsou spočítány numerickou integrací pomocí <a
class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> (text je v angličtině)
nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Vrátit vektor koeficientů kosinové
Fourierovy řady funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovano
u na <strong class="userinput"><code>[0,L]</code></strong>, provedeme sudé periodické rozšíření a spočteme
Fourierovu řadu, která má pouze kosinové členy. Řada je spočítána do <code class="varname">N</code>-té
harmonické. Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.
Poznamenejme, že <strong class="userinput"><code>a@(1)</code></strong> je konstantní koeficient! To znamená,
že <strong class="userinput"><code>a@(n)</code></strong> odkazuje na člen <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>Více informací najdete v encyklopediích <a
class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> (text
je v angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.7 a nověj
ší.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Vrátit funkci, která je kosinovou
Fourierovu řadou funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme sudé periodické rozšíření a spočteme Fourierovu
řadu, která má pouze kosinové členy. Řada je spočítána do <code class="varname">N</code>-té harmonické.
Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada"; target="_top">Wikipedia</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Vrátit vektor koeficientů sinové
Fourierovy řady funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme liché periodické rozšíření a spočteme Fourierovu
řadu, která má pouze sinové členy. Řada je spočítána do <code class="varname">N</code>-té harmonické.
Koeficienty jsou spočítány numerickou integrací pomocí <a class="link" href="ch11s11.html#gel-function-Numeric
alIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada"; target="_top">Wikipedia</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Vrátit funkci, která je sinovou
Fourierovu řadou funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme liché periodické rozšíření a spočteme Fourierovu
řadu, která má pouze sinové členy. Řada je spo
čítána do <code class="varname">N</code>-té harmonické. Koeficienty jsou spočítány numerickou integrací
pomocí <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> (text je v angličtině)
nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Fourierova_%C5%99ada";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integrovat pravidlem nastaveným v
NumericalIntegralFunction jako funkcí f od a do b pomocí kroků NumericalIntegralSteps.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsi
s">NumericalLeftDerivative (f,x0)</pre><p>Zkusit vypočítat numerickou levou derivaci.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,serie_pro_uspech,N)</pre><p>Pokusit se
spočítat limitu f(step_fun(i)) pro i od 1 do N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Zkusit vypočítat numerickou pravou
derivaci.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Vrátit funkci, která je lichým periodickým rozšířením
<code class="function">f</code> s poloviční periodou <code class="varname">L</code>. Tj. funkce definovaná na
intervalu <strong class="userinput"><cod
e>[0,L]</code></strong> rozšířená, aby byla lichá na <strong class="userinput"><code>[-L,L]</code></strong>
a pak rozšířená, aby byla periodická s periodou <strong
class="userinput"><code>2*L</code></strong>.</p><p>Viz také <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> a <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Spočítat jednostrannou derivaci pomocí
pětibodového vzorce.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Spočítat jednostrannou derivaci pomocí
tříbodového vzorce.</p></dd><dt><span class="t
erm"><a name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Vrátit funkci, která je periodickým rozšířením <code
class="function">f</code> definované na intervalu <strong class="userinput"><code>[a,b]</code></strong> a s
periodou <strong class="userinput"><code>b-a</code></strong>.</p><p>Viz také <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> a <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre
class="synopsis">RightLimit (f,x0)</pre><p>Spočítat limitu zprava funkce reálné proměnné v
x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0
,h)</pre><p>Spočítat oboustrannou derivaci pomocí pětibodového vzorce.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Spočítat oboustrannou derivaci pomocí tříbodového
vzorce.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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lass="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-calculus"></a>Diferenciální/integrální počet </h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integrovat f složeným Simpsonovým pravidlem na
intervalu [a,b] s n podintervaly s chybou podle max(f'''')*h^4*(b-a)/180. Upozorňujeme, že n by mělo být
sudé.</p><p>Více informací najdete v encyklopedii <a class="ulink" href="http://planetmath.org/SimpsonsRule";
target="_top">Planetmath</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance (f,a,b,omezeni_ctvrte_derivace,tolerance)</pre><p>Integrovat
f složeným Simpson
ovým pravidlem na intervalu [a,b] s počtem kroků počítaným podle omezení čtvrté derivace a podle požadované
tolerance.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Zkusit spočítat derivaci, nejprve symbolicky a pak numericky.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="https://cs.wikipedia.org/wiki/Derivace";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Vrátit funkci, která je sudým periodickým rozšířením
<code class="function">f</code> s poloviční periodou <code class="varname">L</code>. Tj. funkce definovaná na
inter
valu <strong class="userinput"><code>[0,L]</code></strong> rozšířená, aby byla sudá na <strong
class="userinput"><code>[-L,L]</code></strong> a pak rozšířená, aby byla periodická s periodou <strong
class="userinput"><code>2*L</code></strong>.</p><p>Viz také <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> a <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Vrátit funkci, která je Fourierovu řadou s koeficienty
danými vektory <code class="varname">a</code> (sinové) a <code class="varname">b</code> (kosinové). Vezměte
na vědomí, že <strong class="userinput"><code>a@(1)</code></strong> je konstantní koeficient! To znamená, že
<strong class="userinput"><code>a@(n)</
code></strong> odkazuje na člen <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, zatímco
<strong class="userinput"><code>b@(n)</code></strong> odkazuje na člen <strong
class="userinput"><code>sin(x*n*pi/L)</code></strong>. Buďto <code class="varname">a</code> nebo <code
class="varname">b</code> může být <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(fce,start,prirustek)</pre><p>Zkusit spočítat nekonečný součin funkce s jedním parametrem.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (fce,arg,start,prirustek)</pre><p>Zkusit spočítat nekonečný součin funkce
se dvěma parametry s fce (arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(fce,start,prirustek)</pre><p>Zkusit spočítat nekonečný součet funkce s jedním parametrem.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (fce,arg,start,prirustek)</pre><p>Zkusit spočítat nekonečný součet funkce se
dvěma parametry s fce (arg,n).</p
</dd><dt><span class="term"><a name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre
class="synopsis">IsContinuous (f,x0)</pre><p>Zkusit zjistit pomocí výpočtu limity v x0, jestli je funkce
reálné proměnné v tomto bodě spojitá.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Otestovat na diferencovatelnost aproximací limit zleva a
zprava a porovnáním.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Spočítat limitu zleva funkce reálné proměnné v x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Spočítat
limitu funkce reálné proměnné v x0. Zkusí vypočítat limitu zleva i zprava.</p></dd><dt><span
class="term"><a name="gel-function-MidpointRule"></a>Midp
ointRule</span></dt><dd><pre class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integrovat trojúhelníkovou
metodou (pravidlem prostředního bodu).</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alternativní názvy: <code
class="function">NDerivative</code></p><p>Zkusit vypočítat numerickou derivaci.</p><p>Více informací najdete
v encyklopedii <a class="ulink" href="https://cs.wikipedia.org/wiki/Derivace";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Vrátit vektor vektorů <strong
class="userinput"><code>[a,b]</code></strong>, kde <code class="varname">a</code> jsou kosinové koeficienty a
<code class="varname">b</code> sinové koeficienty Fourierovy řa
dy funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code> (tj.
definovanou na <strong class="userinput"><code>[-L,L]</code></strong> a periodicky rozšířenou) s numericky
spočítanými koeficienty do <code class="varname">N</code>-té harmonické. Koeficienty jsou spočítány
numerickou integrací pomocí <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Vrátit funkci, která je Fourierovou řadou
funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code> (tj. definovanou
na <strong class="userinput"><code>[-L,L]</code></strong> a periodicky rozšířenou) s numericky spočítanými
koeficienty do <code class="varname">N</code>-té harmonické. Jde o čistě trigonometrickou řadu složenou ze
sinů a kosinů. Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Vrátit vektor koeficientů kosinové
Fourierovy řady funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme sudé periodické rozšíření a spočteme Fourierovu
řadu, která má pouze kosinové členy. Řada je spočítána do <code class="varname">N</code>-té harmonické.
Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.
Poznamenejme, že <strong class="userinput"><code>a@(1)</code></strong> je k
onstantní koeficient! To znamená, že <strong class="userinput"><code>a@(n)</code></strong> odkazuje na člen
<strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Vrátit funkci, která je kosinovou
Fourierovu řadou funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme sudé periodické rozšíření a spočteme Fourierovu
řadu, která má pouze kosinové členy. Řada je spočítána do <code class="varname">N</code>-té harmonické.
Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Vrátit vektor koeficientů sinové
Fourierovy řady funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme liché periodické rozšíření a spočteme Fourierovu
řadu, která má pouze sinové členy. Řada je spočítána do <code class="varname">N</code>-té harmonické.
Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Vrátit funkci, která je sinovou
Fourierovu řadou funkce <code class="function">f</code> s poloviční periodou <code class="varname">L</code>.
To jest, vezmeme funkci <code class="function">f</code> definovanou na <strong
class="userinput"><code>[0,L]</code></strong>, provedeme liché periodické rozšíření a spočteme Fourierovu
řadu, která má pouze sinové členy. Řada je spočítána do <code class="varname">N</code>-té harmonické.
Koeficienty jsou spočítány numerickou integrací pomocí <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Verze 1.0.7 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integrovat pravidlem nastaveným v
NumericalIntegralFunction jako funkcí f od a do b pomocí kroků NumericalIntegralSteps.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Zkusit vypočítat numerickou levou
derivaci.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,serie_pro_uspech,N)</pre><p>Pokusit se
spočítat limitu f(step_fun(i)) pro i od 1 do N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">N
umericalRightDerivative (f,x0)</pre><p>Zkusit vypočítat numerickou pravou derivaci.</p></dd><dt><span
class="term"><a name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Vrátit funkci, která je lichým periodickým rozšířením
<code class="function">f</code> s poloviční periodou <code class="varname">L</code>. Tj. funkce definovaná na
intervalu <strong class="userinput"><code>[0,L]</code></strong> rozšířená, aby byla lichá na <strong
class="userinput"><code>[-L,L]</code></strong> a pak rozšířená, aby byla periodická s periodou <strong
class="userinput"><code>2*L</code></strong>.</p><p>Viz také <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> a <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a name="gel-function-OneSide
dFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Spočítat jednostrannou derivaci pomocí
pětibodového vzorce.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Spočítat jednostrannou derivaci pomocí
tříbodového vzorce.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Vrátit funkci, která je periodickým rozšířením <code
class="function">f</code> definované na intervalu <strong class="userinput"><code>[a,b]</code></strong> a s
periodou <strong class="userinput"><code>b-a</code></strong>.</p><p>Viz také <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> a <a class="link"
href="ch11s11.htm
l#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>Verze 1.0.7 a
novější.</p></dd><dt><span class="term"><a name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre
class="synopsis">RightLimit (f,x0)</pre><p>Spočítat limitu zprava funkce reálné proměnné v
x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Spočítat oboustrannou derivaci pomocí pětibodového
vzorce.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Spočítat oboustrannou derivaci pomocí tříbodového
vzorce.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s10.html">Předcházejíc�
�</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch11s12.html">Další</a></td></tr><tr><td width="40%" align="left"
valign="top">Kombinatorika </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Domů</a></td><td width="40%" align="right" valign="top">
Funkce</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s12.html b/help/cs/html/ch11s12.html
index dbe49ff..cba1456 100644
--- a/help/cs/html/ch11s12.html
+++ b/help/cs/html/ch11s12.html
@@ -1 +1,59 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Funkce</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s11.html" title="Diferenciální/integrální počet"><link rel="next" href="ch11s13.html" title="Řešení
rovnic"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Funkce</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s13.html">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div>
<h2 class="title" style="clear: both"><a
name="genius-gel-function-list-functions"></a>Funkce</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Alternativní názvy: <code class="function">Arg</code><code
class="function">arg</code></p><p>Argument (orientovaný úhel) komplexního čísla.</p></dd><dt><span
class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0
(x)</pre><p>Besselova funkce prvního druhu řádu 0. Je implementována pouze pro reálná čísla.</p><p>Více
informací najdete v encyklopedii <a class="ulink" href="http://cs.wikipedia.org/wiki/Besselova_funkce";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Besselova
funk
ce prvního druhu řádu 1. Je implementována pouze pro reálná čísla.</p><p>Více informací najdete v
encyklopedii <a class="ulink" href="http://cs.wikipedia.org/wiki/Besselova_funkce";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn
(n,x)</pre><p>Besselova funkce prvního druhu řádu <code class="varname">n</code>. Je implementována pouze pro
reálná čísla.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Besselova_funkce"; target="_top">Wikipedia</a>.</p><p>Verze 1.0.16 a
novější.</p></dd><dt><span class="term"><a name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre
class="synopsis">BesselY0 (x)</pre><p>Besselova funkce druhého druhu řádu 0. Je implementována pouze pro
reálná čísla.</p><p>Více informací najdete v encyklopedii <a class="ulink" href="http://cs.wi
kipedia.org/wiki/Besselova_funkce" target="_top">Wikipedia</a>.</p><p>Verze 1.0.16 a
novější.</p></dd><dt><span class="term"><a name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre
class="synopsis">BesselY1 (x)</pre><p>Besselova funkce druhého druhu řádu 1. Je implementována pouze pro
reálná čísla.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Besselova_funkce"; target="_top">Wikipedia</a>.</p><p>Verze 1.0.16 a
novější.</p></dd><dt><span class="term"><a name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre
class="synopsis">BesselYn (n,x)</pre><p>Besselova funkce druhého druhu řádu <code class="varname">n</code>.
Je implementována pouze pro reálná čísla.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Besselova_funkce"; target="_top">Wikipedia</a>.</p><p>Verze 1.0.16 a
novější.</p></dd><dt><span class="term"><a name="gel-function-Dirich
letKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichletovo jádro řádu <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Vrátit 1, když a jen když jsou všechny prvky nulové.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alternativní názvy: <code class="function">erf</code></p><p>Chybová funkce, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/ErrorFunction"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="https://cs.wikipedia.org/wiki/Chybov%C3%A1_funkce";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-FejerKernel"></a>FejerKernel</span></dt><d
d><pre class="synopsis">FejerKernel (n,t)</pre><p>Fejerovo jádro řádu <code class="varname">n</code>
vyhodnocené v <code class="varname">t</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/FejerKernel"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alternativní názvy: <code class="function">Gamma</code></p><p>Funkce Gama. V současnosti je
implementována pouze pro reálná čísla.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Gama_funkce"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre
class="synopsis">Kroneck
erDelta (v)</pre><p>Vrátit 1, když a jen když se všechny prvky rovnají.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>Hlavní
větev Lambertovy funkce W vypočítaná pro čistě reálná čísla větší nebo rovna <strong
class="userinput"><code>-1/e</code></strong>. Funkce <code class="function">LambertW</code> je inverzní k
výrazu <strong class="userinput"><code>x*e^x</code></strong>. Dokonce i pro reálná <code
class="varname">x</code> tento výraz není jedna k jedné a proto má dvě větve pro <strong
class="userinput"><code>[-1/e,0)</code></strong>. Viz <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> ohledně další reálné
větve.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://en.wikipedia.org/wiki/Lambert_W_function"; target="_top">Wikipedia</a> (text je v
angličtině).</p><p>Verze 1
.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1
(x)</pre><p>Vedlejší (mínus první) větev Lambertovy funkce W vypočítaná pro čistě reálná čísla větší nebo
rovna <strong class="userinput"><code>-1/e</code></strong>. Funkce <code class="function">LambertWm1</code>
je druhou větví k inverzi výrazu <strong class="userinput"><code>x*e^x</code></strong>. Viz <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> ohledně hlavní
větve.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://en.wikipedia.org/wiki/Lambert_W_function"; target="_top">Wikipedia</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (fce,x,prirust)</pre><p>Najít první hodnotu, kdy f(x)=0.</p></dd><dt>
<span class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Möbiova transformace (lineární lomené zobrazení) kruhu na
sebe sama ku 0.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Möbiova transformace (lineární lomené zobrazení) pomocí dvojpoměrů z2,z3,z4 ku 1,0 a
nekonečnu.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">Moebi
usMappingInftyToInfty (z,z2,z3)</pre><p>Möbiova transformace (lineární lomené zobrazení) pomocí dvojpoměrů
nekonečna ku nekonečnu a z2,z3 ku 1 a 0.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Möbiova transformace (lineární lomené zobrazení)
pomocí dvojpoměrů nekonečna ku 1 a z3,z4 ku 0 a nekonečnu.</p><p>Více informací najdete v encyklopedii <a
class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><
p>Möbiova transformace (lineární lomené zobrazení) pomocí dvojpoměrů nekonečna ku 0 a z2,z4 ku 1 a
nekonečnu.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poissonovo jádro na D(0,1) (nenormalizované na 1, tj. integrál je 2pi).</p></dd><dt><span
class="term"><a name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poissonovo jádro na D(0,R) (nenormalizované na
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Alternativní názvy: <code
class="function">zeta</code></p><p>Riemannova funkce zeta. V současnosti
je implementována jen pro reálná čísla.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://cs.wikipedia.org/wiki/Riemannova_funkce_zeta";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre class="synopsis">UnitStep (x)</pre><p>Funkce
jednotkového skoku je rovna 0 pro x<0 a jedné v ostatních případech. Jedná se o integrál Diracovy funkce
delta. Bývá také nazývána Heavisideova funkce.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Heavisideova_funkce"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre class="synopsis">cis (x)</pre><p>Funkce
<code class="function">cis</code>, což je to stejné jako <strong class="userinput"><
code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Převést
stupně na radiány.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Převést
radiány na stupně.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Vypočítat nenormalizovanou funkci sinc, což je <strong
class="userinput"><code>sin(x)/x</code></strong>. Jestli chcete normalizovanou funkci, volejte <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>Více informací najdete v encyklopedii <a
class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> (článek je v
angličtině).</p><p>Verze 1.0.16 a novější.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation foote
r"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s11.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s13.html">Další</a></td></tr><tr><td width="40%" align="left"
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href="index.html">Domů</a></td><td width="40%" align="right" valign="top"> Řešení
rovnic</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Funkce</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s11.html" title="Diferenciální/integrální počet"><link rel="next" href="ch11s13.html" title="Řešení
rovnic"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Funkce</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s13.html">Další</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div>
<h2 class="title" style="clear: both"><a
name="genius-gel-function-list-functions"></a>Funkce</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Alternativní názvy: <code class="function">Arg</code><code
class="function">arg</code></p><p>Argument (orientovaný úhel) komplexního čísla.</p></dd><dt><span
class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0
(x)</pre><p>Besselova funkce prvního druhu řádu 0. Je implementována pouze pro reálná čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Besselova
funkce prvního druhu řádu 1. Je implementována pouze pro reálná čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn
(n,x)</pre><p>Besselova funkce prvního druhu řádu <code class="varname">n</code>. Je implementována pouze pro
reálná čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Besselova
funkce druhého druhu řádu 0. Je implementována pouze pro reálná čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Besselova
funkce druhého druhu řádu 1. Je implementována pouze pro reálná čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn
(n,x)</pre><p>Besselova funkce druhého druhu řádu <code class="varname">n</code>. Je implementována pouze pro
reálná čísla.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichletovo jádro řádu <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Vrátit 1, když a jen když jsou všechny prvky nulové.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alternativní názvy: <code class="function">erf</code></p><p>Chybová funkce, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/ErrorFunction"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="https://cs.wikipedia.org/wiki/Chybov%C3%A1_funkce"; target="_top">Wikipedia</a>.
</p></dd><dt><span class="term"><a name="gel-function-FejerKernel"></a>FejerKernel</span></dt><dd><pre
class="synopsis">FejerKernel (n,t)</pre><p>Fejerovo jádro řádu <code class="varname">n</code> vyhodnocené v
<code class="varname">t</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/FejerKernel"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alternativní názvy: <code class="function">Gamma</code></p><p>Funkce Gama. V současnosti je
implementována pouze pro reálná čísla.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Vrátit 1, když a jen když se všechny prvky rovnají.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>Hlavní
větev Lambertovy funkce W vypočítaná pro čistě reálná čísla větší nebo rovna <strong
class="userinput"><code>-1/e</code></strong>. Funkce <code class="function">LambertW</code> je inverzní k
výrazu <strong class="userinput"><code>x*e^x</code></strong>. Dokonce i pro reálná <code
class="varname">x</code> tento výraz není jedna k jedné a proto má dvě větve pro <strong
class="userinput"><code>[-1/e,0)</code></strong>. Viz <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> ohledně další reálné
větve.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1
(x)</pre><p>Vedlejší (mínus první) větev Lambertovy funkce W vypočítaná pro čistě reálná čísla větší nebo
rovna <strong class="userinput"><code>-1/e</code></strong>. Funkce <code class="function">LambertWm1</code>
je druhou větví k inverzi výrazu <strong class="userinput"><code>x*e^x</code></strong>. Viz <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> ohledně hlavní
větve.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (fce,x,prirust)</pre><p>Najít první hodnotu, kdy f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Möbiova transformace (lineární lomené zobrazení) kruhu na
sebe sama ku 0.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Möbiova transformace (lineární lomené zobrazení) pomocí dvojpoměrů z2,z3,z4 ku 1,0 a
nekonečnu.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Möbiova transformace (lineární lomené
zobrazení) pomocí dvojpoměrů nekonečna ku nekonečnu a z2,z3 ku 1 a 0.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Möbiova transformace (lineární lomené zobrazení)
pomocí dvojpoměrů nekonečna ku 1 a z3,z4 ku 0 a nekonečnu.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Möbiova transformace (lineární lomené zobrazení)
pomocí dvojpoměrů nekonečna ku 0 a z2,z4 ku 1 a nekonečnu.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poissonovo jádro na D(0,1) (nenormalizované na 1, tj. integrál je 2pi).</p></dd><dt><span
class="term"><a name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poissonovo jádro na D(0,R) (nenormalizované na
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Alternativní názvy: <code
class="function">zeta</code></p><p>Riemannova funkce zeta. V současnosti je implementována jen pro reálná
čísla.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>Funkce jednotkového skoku je rovna 0 pro x<0 a jedné v ostatních
případech. Jedná se o integrál Diracovy funkce delta. Bývá také nazývána Heavisideova funkce.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>Funkce <code class="function">cis</code>, což je to stejné jako <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Převést
stupně na radiány.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Převést
radiány na stupně.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Vypočítat nenormalizovanou funkci sinc, což je <strong
class="userinput"><code>sin(x)/x</code></strong>. Jestli chcete normalizovanou funkci, volejte <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ </p><p>Verze 1.0.16 a novější.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Diferenciální/integrální
počet </td><td width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%"
align="right" valign="top"> Řešení rovnic</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s13.html b/help/cs/html/ch11s13.html
index 83027cc..ed518b1 100644
--- a/help/cs/html/ch11s13.html
+++ b/help/cs/html/ch11s13.html
@@ -1,4 +1,27 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Řešení
rovnic</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11.
Seznam funkcí GEL"><link rel="prev" href="ch11s12.html" title="Funkce"><link rel="next" href="ch11s14.html"
title="Statistika"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Řešení rovnic</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s12.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s14.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="titl
e" style="clear: both"><a name="genius-gel-function-list-equation-solving"></a>Řešení
rovnic</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre class="synopsis">CubicFormula
(p)</pre><p>Vypočítat kořeny kubického (3. stupně) polynomu pomocí kubické rovnice. Polynom by měl být zadán
jako vektor koeficientů. Tj. <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> odpovídá vektoru
<strong class="userinput"><code>[1,2,0,4]</code></strong>. Vrací sloupcový vektor tří řešení. První řešení je
vždy reálné, protože kubická rovnice má vždy jedno reálné řešení.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a> (text
je v angličtině), <a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a> (
text je v angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Kubick%C3%A1_rovnice";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>Použít klasickou Eulerovu metodu k numerickému řešení y'=f(x,y) pro počáteční <code
class="varname">x0</code>, <code class="varname">y0</code> měnící se do <code class="varname">x1</code> s
přírůstky <code class="varname">n</code> a vrátit <code class="varname">y</code> v <code
class="varname">x1</code>. Pokud nechcete výslovně použít Eulerovu metodu, měli byste vážně popřemýšlet o
použití <a class="link" href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a> k řešení obyčejných
diferenciálních rovnic.</p><p>Systémy je možné vyřešit jednoduše tak, že <code class="varname">y</code> musí
být všude (sloupcový) vektor. To znamená, že <code class
="varname">y0</code> může být vektor v případech, kdy by <code class="varname">f</code> mělo přebírat <code
class="varname">x</code> a vektor stejné velikosti pro druhý argument a mělo by vracet vektor stejné
velikosti.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/EulerForwardMethod.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Eulerova_metoda";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>Použít klasickou Eulerovu metodu k numerickému
řešení y'=f(x,y) pro počáteční <code class="varname">x0</code>, <code class="varname">y0</code> měnící se do
<code class="varname">x1</code> s přírůstky <code class="varname">n</code> a vrátit matici 2 krát <strong
class="userinput"><code>n+1</
code></strong> s hodnotami <code class="varname">x</code> a <code class="varname">y</code>. Pokud nechcete
výslovně použít Eulerovu metodu, měli byste vážně popřemýšlet o použití <a class="link"
href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a> k řešení obyčejných diferenciálních
rovnic. Vhodné pro zapojení do <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> nebo <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Příklad: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Řešení
rovnic</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Příručka k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11.
Seznam funkcí GEL"><link rel="prev" href="ch11s12.html" title="Funkce"><link rel="next" href="ch11s14.html"
title="Statistika"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Řešení rovnic</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s12.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s14.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="titl
e" style="clear: both"><a name="genius-gel-function-list-equation-solving"></a>Řešení
rovnic</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre class="synopsis">CubicFormula
(p)</pre><p>Vypočítat kořeny kubického (3. stupně) polynomu pomocí kubické rovnice. Polynom by měl být zadán
jako vektor koeficientů. Tj. <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> odpovídá vektoru
<strong class="userinput"><code>[1,2,0,4]</code></strong>. Vrací sloupcový vektor tří řešení. První řešení je
vždy reálné, protože kubická rovnice má vždy jedno reálné řešení.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>Použít klasickou Eulerovu metodu k numerickému řešení y'=f(x,y) pro počáteční <code
class="varname">x0</code>, <code class="varname">y0</code> měnící se do <code class="varname">x1</code> s
přírůstky <code class="varname">n</code> a vrátit <code class="varname">y</code> v <code
class="varname">x1</code>. Pokud nechcete výslovně použít Eulerovu metodu, měli byste vážně popřemýšlet o
použití <a class="link" href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a> k řešení obyčejných
diferenciálních rovnic.</p><p>Systémy je možné vyřešit jednoduše tak, že <code class="varname">y</code> musí
být všude (sloupcový) vektor. To znamená, že <code class="varname">y0</code> může být vektor v případech, kdy
by <code class="varname">f</code> mělo přebírat <code class="va
rname">x</code> a vektor stejné velikosti pro druhý argument a mělo by vracet vektor stejné velikosti.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
+ Use classical Euler's method to numerically solve y'=f(x,y) for
+ initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
+ <code class="varname">x1</code> with <code class="varname">n</code> increments,
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
+ <code class="varname">x</code> and <code class="varname">y</code> values.
+ Unless you explicitly want to use Euler's method, you should really
+ think about using
+ <a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a>
+ for solving ODE.
+ Suitable
+ for plugging into
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
+ </p><p>Příklad: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line =
EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponenciální
vývoj");</code></strong>
</pre><p>Systémy je možné vyřešit jednoduše tak, že <code class="varname">y</code> musí být všude
(sloupcový) vektor. To znamená, že <code class="varname">y0</code> může být vektor v případech, kdy by <code
class="varname">f</code> mělo přebírat <code class="varname">x</code> a vektor stejné velikosti pro druhý
argument a mělo by vracet vektor stejné velikosti.</p><p>Výstup pro systém je nicméně matice n krát 2 s
druhou položkou v podobě vektoru. Když si přejete vykreslit čáru, ujistěte se, že používáte řádkové vektory a
pak převeďte matici na vektor pomocí <a class="link"
href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a> a vyberte si pravý sloupec. Například:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
@@ -9,9 +32,44 @@
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","První");</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Druhý");</code></strong>
-</pre><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/EulerForwardMethod.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://cs.wikipedia.org/wiki/Eulerova_metoda";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.10 a novější.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Najít kořen funkce pomocí metody bisekce. <code
class="varname">a</code> a <code class="varname">b</code> je počáteční odhad intervalu, <strong
class="userinput"><code>f(a)</code></strong> a <strong class="userinput"><code>f(b)</code></strong> by měly
mít opačná znaménka. <code class="varname">TOL</code> je požadovaná tolerance a <code
class="varname">N</code> je omezení počtu iterací, které mají proběhnout, 0 značí bez omezení. Funkce vrací
vektor <strong class="userinput">
<code>[uspech,hodnota,iteratce]</code></strong>, kde <code class="varname">uspech</code> je pravdivostní
hodnota signalizující úspěch, <code class="varname">hodnota</code> je poslední spočtená hodnota a <code
class="varname">iterace</code> je počet dokončených iterací.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Najít kořen funkce pomocí metody tětiv. <code
class="varname">a</code> a <code class="varname">b</code> je počáteční odhad intervalu, <strong
class="userinput"><code>f(a)</code></strong> a <strong class="userinput"><code>f(b)</code></strong> by měly
mít opačná znaménka. <code class="varname">TOL</code> je požadovaná tolerance a <code
class="varname">N</code> je omezení počtu iterací, které mají proběhnout, 0 značí bez omezení. Funkce vrací
vektor <strong class="userinput"><code>[uspech,hodnota,
iteratce]</code></strong>, kde <code class="varname">uspech</code> je pravdivostní hodnota signalizující
úspěch, <code class="varname">hodnota</code> je poslední spočtená hodnota a <code
class="varname">iterace</code> je počet dokončených iterací.</p></dd><dt><span class="term"><a
name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Najít kořen funkce pomocí Mullerovy metody.
<code class="varname">TOL</code> je požadovaná tolerance a <code class="varname">N</code> je omezení počtu
iterací, které mají proběhnout, 0 značí bez omezení. Funkce vrací vektor <strong
class="userinput"><code>[uspech,hodnota,iteratce]</code></strong>, kde <code class="varname">uspech</code> je
pravdivostní hodnota signalizující úspěch, <code class="varname">hodnota</code> je poslední spočtená hodnota
a <code class="varname">iterace</code> je počet dokončených iter
ací.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Najít kořen funkce pomocí metody sečen. <code class="varname">a</code> a <code
class="varname">b</code> je počáteční odhad intervalu, <strong class="userinput"><code>f(a)</code></strong> a
<strong class="userinput"><code>f(b)</code></strong> by měly mít opačná znaménka. <code
class="varname">TOL</code> je požadovaná tolerance a <code class="varname">N</code> je omezení počtu iterací,
které mají proběhnout, 0 značí bez omezení. Funkce vrací vektor <strong
class="userinput"><code>[uspech,hodnota,iteratce]</code></strong>, kde <code class="varname">uspech</code> je
pravdivostní hodnota signalizující úspěch, <code class="varname">hodnota</code> je poslední spočtená hodnota
a <code class="varname">iterace</code> je počet dokončených iterací.</p></dd><dt><span class="term"><a nam
e="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,odhad,epsilon,maxn)</pre><p>Najde nuly pomocí Halleyovy metody. <code class="varname">f</code> je
funkce, <code class="varname">df</code> je její derivace a <code class="varname">ddf</code> její druhá
derivace. <code class="varname">odhad</code> je počáteční odhad. Funkce vrací výsledek po dvou úspěšných
hodnotách, které každá spadají do <code class="varname">epsilon</code> nebo po <code
class="varname">maxn</code> pokusech, v kterémžto případě vrací <code class="constant">null</code>, což značí
selhání.</p><p>Viz také <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a> a <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Příklad vyhledání druhé odmocniny z 10: </p><pre
class="screen"><code class="
prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
-</pre><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://en.wikipedia.org/wiki/Halley%27s_method"; target="_top">Wikipedia</a> (text je v
angličtině).</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,odhad,epsilon,maxn)</pre><p>Najde nuly pomocí metody tečen (Newtonovy metody). <code
class="varname">f</code> je funkce a <code class="varname">df</code> je její derivace. <code
class="varname">odhad</code> je počáteční odhad. Funkce vrací výsledek po dvou úspěšných hodnotách, které
každá spadají do <code class="varname">epsilon</code> nebo po <code class="varname">maxn</code> pokusech, v
kterémžto případě vrací <code class="constant">null</code>, což značí selhání.</p><p>Viz také <a class="link"
href="ch11s15.html#gel-function-NewtonsMethodPoly"><code class="function">NewtonsMethodPoly</code></a> a <a
class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Příklad vyhledání druhé odmocniny z 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
-</pre><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Metoda_te%C4%8Den"; target="_top">Wikipedia</a>.</p><p>Verze 1.0.18 a
novější.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Vypočítat kořeny polynomu (1. až 4. stupně) pomocí jedné z rovnic pro takovéto polynomy. Polynom
by měl být zadán jako vektor koeficientů. Tj. <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong>
odpovídá vektoru <strong class="userinput"><code>[1,2,0,4]</code></strong>. Vrací sloupcový vektor
řešení.</p><p>Funkce volá <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> a <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-fun
ction-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre class="synopsis">QuadraticFormula
(p)</pre><p>Vypočítat kořeny kvadratického (2. stupně) polynomu pomocí kvadratické rovnice. Polynom by měl
být zadán jako vektor koeficientů. Tj. <strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong>
odpovídá vektoru <strong class="userinput"><code>[1,2,3]</code></strong>. Vrací sloupcový vektor dvou
řešení.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> (text je v angličtině) a <a
class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html"; target="_top">Mathworld</a> (text je
v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Vypočítat kořeny kvartického (4. stupně) polynomu pomocí kvartické rovnice. Polynom by měl být
zadán jako vektor koeficientů. Tj. <strong class="userinput"><code>5*x^4 + 2*x + 1</code></strong> odpovídá
vektoru <strong class="userinput"><code>[1,2,0,0,5]</code></strong>. Vrací sloupcový vektor čtyř
řešení.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a> (text je v angličtině), <a
class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html"; target="_top">Mathworld</a> (text je v
angličtině) a <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Použít klasickou neadaptivní Rungeho-Kuttovu metodu čtvrtého řádu k numerickému řešení
y'=f(x,y) pro počáteční <code class="varname">x0</code>, <code class="varname">y0</code> měn
ící se do <code class="varname">x1</code> s přírůstky <code class="varname">n</code>, vrací <code
class="varname">y</code> v <code class="varname">x1</code>.</p><p>Systémy je možné vyřešit jednoduše tak, že
<code class="varname">y</code> musí být všude (sloupcový) vektor. To znamená, že <code
class="varname">y0</code> může být vektor v případech, kdy by <code class="varname">f</code> mělo přebírat
<code class="varname">x</code> a vektor stejné velikosti pro druhý argument a mělo by vracet vektor stejné
velikosti.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/Runge-KuttaMethod.html"; target="_top">Mathworld</a> (text je v angličtině)
a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Numerick%C3%A9_%C5%99e%C5%A1en%C3%AD_oby%C4%8Dejn%C3%BDch_diferenci%C3%A1ln%C3%ADch_rovnic#Metody_Runge-Kutta";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a name="gel-function-RungeKuttaFull
"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull (f,x0,y0,x1,n)</pre><p>Použít
klasickou neadaptivní metodu Runge-Kutta čtvrtého řádu k numerickému řešení y'=f(x,y) pro počáteční <code
class="varname">x0</code>, <code class="varname">y0</code> měnící se do <code class="varname">x1</code> s
přírůstky <code class="varname">n</code> a vrátit matici 2 krát <strong
class="userinput"><code>n+1</code></strong> s hodnotami <code class="varname">x</code> a <code
class="varname">y</code>. Vhodné pro zapojení do <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> nebo <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Příklad: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Verze 1.0.10 a novější.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Najít kořen funkce pomocí metody bisekce. <code
class="varname">a</code> a <code class="varname">b</code> je počáteční odhad intervalu, <strong
class="userinput"><code>f(a)</code></strong> a <strong class="userinput"><code>f(b)</code></strong> by měly
mít opačná znaménka. <code class="varname">TOL</code> je požadovaná tolerance a <code
class="varname">N</code> je omezení počtu iterací, které mají proběhnout, 0 značí bez omezení. Funkce vrací
vektor <strong class="userinput"><code>[uspech,hodnota,iteratce]</code></strong>, kde <code
class="varname">uspech</code> je pravdivostní hodnota signalizující úspěch, <code
class="varname">hodnota</code> je poslední spočtená hodnota a <code class="varname">iterace</code> je počet
dokončených iterac
í.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Najít kořen funkce pomocí metody tětiv. <code
class="varname">a</code> a <code class="varname">b</code> je počáteční odhad intervalu, <strong
class="userinput"><code>f(a)</code></strong> a <strong class="userinput"><code>f(b)</code></strong> by měly
mít opačná znaménka. <code class="varname">TOL</code> je požadovaná tolerance a <code
class="varname">N</code> je omezení počtu iterací, které mají proběhnout, 0 značí bez omezení. Funkce vrací
vektor <strong class="userinput"><code>[uspech,hodnota,iteratce]</code></strong>, kde <code
class="varname">uspech</code> je pravdivostní hodnota signalizující úspěch, <code
class="varname">hodnota</code> je poslední spočtená hodnota a <code class="varname">iterace</code> je počet
dokončených iterací.</p></dd><dt><span
class="term"><a name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Najít kořen funkce pomocí Mullerovy metody.
<code class="varname">TOL</code> je požadovaná tolerance a <code class="varname">N</code> je omezení počtu
iterací, které mají proběhnout, 0 značí bez omezení. Funkce vrací vektor <strong
class="userinput"><code>[uspech,hodnota,iteratce]</code></strong>, kde <code class="varname">uspech</code> je
pravdivostní hodnota signalizující úspěch, <code class="varname">hodnota</code> je poslední spočtená hodnota
a <code class="varname">iterace</code> je počet dokončených iterací.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Najít kořen funkce pomocí metody sečen. <code class="varname">a</code> a <code
class="varname">b</code> je
počáteční odhad intervalu, <strong class="userinput"><code>f(a)</code></strong> a <strong
class="userinput"><code>f(b)</code></strong> by měly mít opačná znaménka. <code class="varname">TOL</code> je
požadovaná tolerance a <code class="varname">N</code> je omezení počtu iterací, které mají proběhnout, 0
značí bez omezení. Funkce vrací vektor <strong
class="userinput"><code>[uspech,hodnota,iteratce]</code></strong>, kde <code class="varname">uspech</code> je
pravdivostní hodnota signalizující úspěch, <code class="varname">hodnota</code> je poslední spočtená hodnota
a <code class="varname">iterace</code> je počet dokončených iterací.</p></dd><dt><span class="term"><a
name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,odhad,epsilon,maxn)</pre><p>Najde nuly pomocí Halleyovy metody. <code class="varname">f</code> je
funkce, <code class="varname">df</code> je její derivace a <code cla
ss="varname">ddf</code> její druhá derivace. <code class="varname">odhad</code> je počáteční odhad. Funkce
vrací výsledek po dvou úspěšných hodnotách, které každá spadají do <code class="varname">epsilon</code> nebo
po <code class="varname">maxn</code> pokusech, v kterémžto případě vrací <code class="constant">null</code>,
což značí selhání.</p><p>Viz také <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a> a <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Příklad vyhledání druhé odmocniny z 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,odhad,epsilon,maxn)</pre><p>Najde nuly pomocí metody tečen (Newtonovy metody). <code
class="varname">f</code> je funkce a <code class="varname">df</code> je její derivace. <code
class="varname">odhad</code> je počáteční odhad. Funkce vrací výsledek po dvou úspěšných hodnotách, které
každá spadají do <code class="varname">epsilon</code> nebo po <code class="varname">maxn</code> pokusech, v
kterémžto případě vrací <code class="constant">null</code>, což značí selhání.</p><p>Viz také <a class="link"
href="ch11s15.html#gel-function-NewtonsMethodPoly"><code class="function">NewtonsMethodPoly</code></a> a <a
class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Příklad vyhledání druhé odmocniny z 10:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Vypočítat kořeny polynomu (1. až 4. stupně) pomocí jedné z rovnic pro takovéto polynomy. Polynom
by měl být zadán jako vektor koeficientů. Tj. <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong>
odpovídá vektoru <strong class="userinput"><code>[1,2,0,4]</code></strong>. Vrací sloupcový vektor
řešení.</p><p>Funkce volá <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> a <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre
class="synopsis">QuadraticFormula (p)</pre><p>Vypočítat kořeny kvadratického
(2. stupně) polynomu pomocí kvadratické rovnice. Polynom by měl být zadán jako vektor koeficientů. Tj.
<strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> odpovídá vektoru <strong
class="userinput"><code>[1,2,3]</code></strong>. Vrací sloupcový vektor dvou řešení.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Vypočítat kořeny kvartického (4. stupně) polynomu pomocí kvartické rovnice. Polynom by měl být
zadán jako vektor koeficientů. Tj. <strong class="userinput"><code>5*x^4 + 2*x + 1</code></strong> odpovídá
vektoru <strong class="userinput"><code>[1,2,0,0,5]</code></strong>. Vrací sloupcový vektor čtyř
řešení.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Použít klasickou neadaptivní Rungeho-Kuttovu metodu čtvrtého řádu k numerickému řešení
y'=f(x,y) pro počáteční <code class="varname">x0</code>, <code class="varname">y0</code> měnící se do <code
class="varname">x1</code> s přírůstky <code class="varname">n</code>, vrací <code class="varname">y</code> v
<code class="varname">x1</code>.</p><p>Systémy je možné vyřešit jednoduše tak, že <code
class="varname">y</code> musí být všude (sloupcový) vektor. To znamená, že <code class="varname">y0</code>
může být vektor v případech, kdy by <code class="varname">f</code> mělo přebírat <code
class="varname">x</code> a vektor stejné velikosti pro druhý argument a mělo by vracet vektor stejné
velikosti.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
+ Use classical non-adaptive fourth order Runge-Kutta method to
+ numerically solve
+ y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
+ going to <code class="varname">x1</code> with <code class="varname">n</code>
+ increments,
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
+ <code class="varname">x</code> and <code class="varname">y</code> values. Suitable
+ for plugging into
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
+ </p><p>Příklad: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line =
RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponenciální
vývoj");</code></strong>
</pre><p>Systémy je možné vyřešit jednoduše tak, že <code class="varname">y</code> musí být všude
(sloupcový) vektor. To znamená, že <code class="varname">y0</code> může být vektor v případech, kdy by <code
class="varname">f</code> mělo přebírat <code class="varname">x</code> a vektor stejné velikosti pro druhý
argument a mělo by vracet vektor stejné velikosti.</p><p>Výstup pro systém je nicméně matice n krát 2 s
druhou položkou v podobě vektoru. Když si přejete vykreslit čáru, ujistěte se, že používáte řádkové vektory a
pak převeďte matici na vektor pomocí <a class="link"
href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a> a vyberte si pravý sloupec. Například:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
@@ -22,4 +80,8 @@
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","První");</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Druhý");</code></strong>
-</pre><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/Runge-KuttaMethod.html"; target="_top">Mathworld</a> (text je v angličtině)
a <a class="ulink"
href="http://cs.wikipedia.org/wiki/Numerick%C3%A9_%C5%99e%C5%A1en%C3%AD_oby%C4%8Dejn%C3%BDch_diferenci%C3%A1ln%C3%ADch_rovnic#Metody_Runge-Kutta";
target="_top">Wikipedia</a>.</p><p>Verze 1.0.10 a novější.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s12.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Funkce </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%" align="right"
valign="top"> Statistika</td></tr></table><
/div></body></html>
+</pre><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ </p><p>Verze 1.0.10 a novější.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Funkce </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%" align="right"
valign="top"> Statistika</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s14.html b/help/cs/html/ch11s14.html
index 833ae6f..a6f3379 100644
--- a/help/cs/html/ch11s14.html
+++ b/help/cs/html/ch11s14.html
@@ -1 +1,26 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistika</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Příručka k aplikaci Genius"><link rel="up"
href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev" href="ch11s13.html" title="Řešení
rovnic"><link rel="next" href="ch11s15.html" title="Polynomy"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Statistika</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a name="genius-gel-function-list-statistics"></a>Statistika</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alternativní
názvy: <code class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Vypočítat průměr z celé matice.</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Aritmetick%C3%BD_pr%C5%AFm%C4%9Br";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integrál Gaussovy funkce od 0 do <code
class="varname">x</code> (oblast pod no
rmální křivkou).</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> (text je v
angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Normalizovaného Gaussova funkce rozdělení (normální křivka).</p><p>Více informací najdete v
encyklopediích <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> (text je v angličtině) nebo <a class="ulink"
href="http://cs.wikipedia.org/wiki/Norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-Median"></a>Median</span></dt><dd><pre class="synopsis">Median (m)</pre><p>Alternati
vní názvy: <code class="function">median</code></p><p>Vypočítat medián z celé matice.</p><p>Více informací
najdete v encyklopedii <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">stdevp</code></p><p>Spočítat standardní odchylku souboru celé matice.</p></dd><dt><span
class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre class="synopsis">RowAverage
(m)</pre><p>Alternativní názvy: <code class="function">RowMean</code></p><p>Vypočítat průměr každého řádku v
matici.</p><p>Více informací najdete v encyklopediích <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> (text
je v angličtině) nebo <a class="ulink" href="http://cs.wikipedia.org/wiki/Aritmetick%C3%BD_pr%C5%AFm%C4%9Br";
target="_top">Wikipedia</a>.</p></dd><dt><span class="term"><a
name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre class="synopsis">RowMedian
(m)</pre><p>Vypočítat medián každého řádku v matici a vrátit sloupcový vektor mediánů.</p><p>Více informací
najdete v encyklopedii <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> (text je v angličtině).</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">rowstdevp</code></p><p>Spočítat standardní odchylku souboru řádků matice a vrátit svislý
vektor.</p></dd><dt><span class="term"><a name="gel-function-RowStandardDeviation"></a>RowStandardDeviation<
/span></dt><dd><pre class="synopsis">RowStandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">rowstdev</code></p><p>Spočítat standardní odchylku řádků matice a vrátit svislý
vektor.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">stdev</code></p><p>Spočítat standardní odchylku celé matice.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s13.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Řešení rovnic </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Dom�
�</a></td><td width="40%" align="right" valign="top"> Polynomy</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistika</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Příručka k aplikaci Genius"><link rel="up"
href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev" href="ch11s13.html" title="Řešení
rovnic"><link rel="next" href="ch11s15.html" title="Polynomy"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Statistika</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam
funkcí GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a name="genius-gel-function-list-statistics"></a>Statistika</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alternativní
názvy: <code class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integrál Gaussovy funkce od 0 do <code
class="varname">x</code> (oblast pod normální křivkou).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Normalizovaného Gaussova funkce rozdělení (normální křivka).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Alternativní názvy: <code class="function">median</code></p><p>Vypočítat
medián z celé matice.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">stdevp</code></p><p>Spočítat standardní odchylku souboru celé matice.</p></dd><dt><span
class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre class="synopsis">RowAverage
(m)</pre><p>Alternativní názvy: <code class="function">RowMean</code></p><p>Calculate average of each row in
a matrix. That is, compute the
+ arithmetic mean.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Vypočítat medián každého řádku v matici a vrátit sloupcový vektor
mediánů.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">rowstdevp</code></p><p>Spočítat standardní odchylku souboru řádků matice a vrátit svislý
vektor.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">rowstdev</code></p><p>Spočítat standardní odchylku řádků matice a vrátit svislý
vektor.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alternativní názvy: <code
class="function">stdev</code></p><p>Spočítat standardní odchylku celé matice.</p></dd></dl></div></div><div
class="navfooter"><hr><
table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Řešení rovnic </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%" align="right"
valign="top"> Polynomy</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s15.html b/help/cs/html/ch11s15.html
index 8a6f316..4f2364a 100644
--- a/help/cs/html/ch11s15.html
+++ b/help/cs/html/ch11s15.html
@@ -1,2 +1,5 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Polynomy</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s14.html" title="Statistika"><link rel="next" href="ch11s16.html" title="Teorie
množin"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Polynomy</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s14.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s16.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style
="clear: both"><a name="genius-gel-function-list-polynomials"></a>Polynomy</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AddPoly"></a>AddPoly</span></dt><dd><pre class="synopsis">AddPoly (p1,p2)</pre><p>Sečíst
dva polynomy (vektory).</p></dd><dt><span class="term"><a
name="gel-function-DividePoly"></a>DividePoly</span></dt><dd><pre class="synopsis">DividePoly
(p,q,&r)</pre><p>Podělit dva polynomy (jako vektory) pomocí dlouhého dělení. Vrátit rozdíl dvou polynomů.
Volitelný argument <code class="varname">r</code> se použije k vrácení zbytku. Zbytek bude mít nižší řád, než
polynom <code class="varname">q</code>.</p><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://planetmath.org/PolynomialLongDivision"; target="_top">Planetmath</a> (text je v
angličtině).</p></dd><dt><span class="term"><a name="gel-function-IsPoly"></a>IsPoly</span></dt><dd><pre
class="synop
sis">IsPoly (p)</pre><p>Zkontrolovat, zda je vektor použitelný jako polynom.</p></dd><dt><span
class="term"><a name="gel-function-MultiplyPoly"></a>MultiplyPoly</span></dt><dd><pre
class="synopsis">MultiplyPoly (p1,p2)</pre><p>Vynásobit dva polynomy (jako vektory).</p></dd><dt><span
class="term"><a name="gel-function-NewtonsMethodPoly"></a>NewtonsMethodPoly</span></dt><dd><pre
class="synopsis">NewtonsMethodPoly (poly,odhad,epsilon,maxn)</pre><p>Najde kořeny polynomu pomocí metody
tečen (Newtonovy metody). <code class="varname">poly</code> je polynom v podobě vektoru a <code
class="varname">odhad</code> je počáteční odhad. Funkce vrací výsledek po dvou úspěšných hodnotách, které
každá spadají do <code class="varname">epsilon</code> nebo po <code class="varname">maxn</code> pokusech, v
kterémžto případě vrací <code class="constant">null</code>, což značí selhání.</p><p>Viz také <a class="link"
href="ch11s13.html#gel-function-NewtonsMethod"><co
de class="function">NewtonsMethod</code></a>.</p><p>Příklad vyhledání druhé odmocniny z 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethodPoly([-10,0,1],3,10^-10,100)</code></strong>
-</pre><p>Více informací najdete v encyklopedii <a class="ulink"
href="http://cs.wikipedia.org/wiki/Metoda_te%C4%8Den"; target="_top">Wikipedia</a>.</p></dd><dt><span
class="term"><a name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Vypočítat druhou derivaci polynomu (jako
vektoru).</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Derivovat polynom (jako vektor).</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Vytvořit funkci z polynomu (jako vektoru).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,prom...)</pre><p>Vytvořit řetězec z polynomu (jako vektoru).</p></dd><dt><span class="term"><a name="gel-
function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Odečíst dva polynomy (jako vektory).</p></dd><dt><span class="term"><a
name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly (p)</pre><p>Odstranit
nuly z polynomu (jako vektoru).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s14.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Statistika </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%" align="right"
valign="top"> Teorie množin</td></tr></table></div></body></html>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Vypočítat druhou derivaci polynomu (jako
vektoru).</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Derivovat polynom (jako vektor).</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Vytvořit funkci z polynomu (jako vektoru).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,prom...)</pre><p>Vytvořit řetězec z polynomu (jako vektoru).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Odečíst dva polynomy (jako vektory).<
/p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Odstranit nuly z polynomu (jako vektoru).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Předcházející</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top">Statistika </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Domů</a></td><td width="40%" align="right"
valign="top"> Teorie množin</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s18.html b/help/cs/html/ch11s18.html
index 354d509..870c50b 100644
--- a/help/cs/html/ch11s18.html
+++ b/help/cs/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Různé</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s17.html" title="Komutativní algebra"><link rel="next" href="ch11s19.html" title="Symbolické
operace"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Různé</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class=
"title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Různé</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vektor)</pre><p>Převést vektor hodnost ASCII na řetězec.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vektor,abeceda)</pre><p>Převést vektor hodnot abecedy (pozic v řetězci
abecedy) počítaných od 0 na řetězec.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(retezec)</pre><p>Převést retezec na vektor hodnot ASCII.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (retezec,abeceda)</pre><p>Př
evést řetězec na vektor hodnot abecedy (pozic v řetězci) počítaných od 0, za neznámé znaky se dosadí
-1.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Další</a></td></tr><tr><td width="40%" align="left"
valign="top">Komutativní algebra </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Domů</a></td><td width="40%" align="right" valign="top"> Symbolické
operace</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Různé</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Příručka
k aplikaci Genius"><link rel="up" href="ch11.html" title="Kapitola 11. Seznam funkcí GEL"><link rel="prev"
href="ch11s17.html" title="Komutativní algebra"><link rel="next" href="ch11s19.html" title="Symbolické
operace"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Různé</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Předcházející</a> </td><th width="60%" align="center">Kapitola 11. Seznam funkcí
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Další</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class=
"title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Různé</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vektor)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vektor,abeceda)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(retezec)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (retezec,abeceda)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Předcházející</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Nahoru</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Další</a></td></tr><tr><td width="40%" align="left"
valign="top">Komutativní algebra </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Domů</a></td><td width="40%" align="right" valign="top"> Symbolické
operace</td></tr></table></div></body></html>
diff --git a/help/cs/html/ch11s20.html b/help/cs/html/ch11s20.html
index c3cff9b..dc380ce 100644
--- a/help/cs/html/ch11s20.html
+++ b/help/cs/html/ch11s20.html
@@ -2,18 +2,24 @@
<code class="prompt">genius></code> <strong
class="userinput"><code>ExportPlot("/složka/soubor","eps")</code></strong>
</pre><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-LinePlot"></a>LinePlot</span></dt><dd><pre class="synopsis">LinePlot
(fce1,fce2,fce3,...)</pre><pre class="synopsis">LinePlot (fce1,fce2,fce3,x1,x2)</pre><pre
class="synopsis">LinePlot (fce1,fce2,fce3,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlot
(fce1,fce2,fce3,[x1,x2])</pre><pre class="synopsis">LinePlot (fce1,fce2,fce3,[x1,x2,y1,y2])</pre><p>Vykreslí
funkci (nebo několik funkcí) v podobě čárového grafu. Prvních (až 10) argumentů jsou funkce, volitelně můžete
zadat meze vykreslovaného okna jako souřadnice <code class="varname">x1</code>, <code
class="varname">x2</code>, <code class="varname">y1</code>, <code class="varname">y2</code>. Pokud žádné meze
nejsou zadány, použijí se aktuálně nastavené meze (viz <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Pokud
nejsou zadány jen meze v ose y, fu
nkce se propočítají a vezme se jejich minimum a maximu.</p><p>Parametr <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
ovládá vykreslování legendy.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>LinePlot(sin,cos)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlot(`(x)=x^2,-1,1,0,1)</code></strong>
-</pre></dd><dt><span class="term"><a name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre
class="synopsis">LinePlotClear ()</pre><p>Zobrazí okno pro vykreslování čar a vymaže funkce a ostatní čáry,
které jsou v něm vykresleny.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (fce,...)</pre><pre class="synopsis">LinePlotCParametric
(fce,t1,t2,tprirust)</pre><pre class="synopsis">LinePlotCParametric
(fce,t1,t2,tprirust,x1,x2,y1,y2)</pre><p>Vykreslit parametrickou funkci komplexní hodnoty v podobě čárového
grafu. Jako první se předává funkce, která vrací <code class="computeroutput">x+iy</code>, následovaná
volitelnými omezeními <strong class="userinput"><code>t1,t2,tprirust</code></strong> pro <code
class="varname">t</code> a pak mezemi v podobě <strong
class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Pokud žádné me
ze nejsou zadány, použijí se aktuálně nastavené meze (viz <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Místo mezí
x a y je možné zadat řetězec "fit" a meze se pak zvolí podle maximálního rozsahu grafu.</p><p>Parametr <a
class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> ovládá vykreslování legendy.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>Vykreslit čáru z <code class="varname">x1</code>,<code class="varname">y1</code> do <code
class="varname">x2</code>,<code class="varname">y2</code>. <code class="varname">x1</code>,<code
class="varname">y1</code>, <code class="varname">x2</code>,<code class="varname">y2</code> může být pro delší
lom
ené čáry nahrazeno maticí <code class="varname">n</code> krát 2. Připadně vektor <code
class="varname">v</code> může být sloupcový vektor komplexních čísel, což je matice <code
class="varname">n</code> krát 1 a jednotlivá komplexní čísla jsou pak považována za body v
rovině.</p><p>Mohou být přidány dodatečné parametry, které určují barvu, tloušťku a šipky čáry a vykreslení
okna nebo legendy. Stačí přidat argument v podobě řetězce <strong
class="userinput"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>, <strong
class="userinput"><code>"arrow"</code></strong> nebo <strong class="userinput"><code>"legend"</code></strong>
a za ním určit barvu, tloušťku, okno jako 4prvkový vektor, typ šipky nebo legendu. (Šipka a okno jsou
podporovány od verze 1.0.6.)</p><p>Pokud je čára považovaná za vyplněný mnohoúhelník, vyplněný dan
ou barvou, můžete zadat argument <strong class="userinput"><code>"filled"</code></strong>. K dispozici od
verze 1.0.22.</p><p>Barva by měla být buď řetězec symbolizující běžným anglickým slovem barvu, kterou
rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
apod. Nebo druhou možností je zadat barvu ve formátu RGB jako <strong
class="userinput"><code>"#rgb"</code></strong>, <strong class="userinput"><code>"#rrggbb"</code></strong>
nebo <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové
soustavy červené, zelené a modré složky barvy. A nakonec třetí možností, od verze 1.0.18, je také určení
barvy vektorem reálných čísel, která představují červenou, zelenou a modrou složku v rozmezí 0 až 1, např.
<strong class="userinput"><code>[1.0,0.5,0.1]</cod
e></strong>.</p><p>Okno by mělo být zadáno buď obvyklým způsobem jako <strong
class="userinput"><code>[x1,x2,y1,y2]</code></strong> nebo alternativně může být použit řetězec <strong
class="userinput"><code>"fit"</code></strong>, v kterémž to případě bude rozsah x určen přesně a rozsah y
bude nastaven s pětiprocentním přesahem křivky.</p><p>Specifikace šipky by měla být <strong
class="userinput"><code>"origin"</code></strong> (počátek), <strong
class="userinput"><code>"end"</code></strong> (konec), <strong class="userinput"><code>"both"</code></strong>
(obojí) nebo <strong class="userinput"><code>"none"</code></strong> (nic).</p><p>A nakonec legenda, která by
měla být zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě, že se legenda
tiskne.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickne
ss",3)</code></strong>
+</pre></dd><dt><span class="term"><a name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre
class="synopsis">LinePlotClear ()</pre><p>Zobrazí okno pro vykreslování čar a vymaže funkce a ostatní čáry,
které jsou v něm vykresleny.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (fce,...)</pre><pre class="synopsis">LinePlotCParametric
(fce,t1,t2,tprirust)</pre><pre class="synopsis">LinePlotCParametric
(fce,t1,t2,tprirust,x1,x2,y1,y2)</pre><p>Vykreslit parametrickou funkci komplexní hodnoty v podobě čárového
grafu. Jako první se předává funkce, která vrací <code class="computeroutput">x+iy</code>, následovaná
volitelnými omezeními <strong class="userinput"><code>t1,t2,tprirust</code></strong> pro <code
class="varname">t</code> a pak mezemi v podobě <strong
class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Pokud žádné me
ze nejsou zadány, použijí se aktuálně nastavené meze (viz <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Místo mezí
x a y je možné zadat řetězec "fit" a meze se pak zvolí podle maximálního rozsahu grafu.</p><p>Parametr <a
class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> ovládá vykreslování legendy.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>Vykreslit čáru z <code class="varname">x1</code>,<code class="varname">y1</code> do <code
class="varname">x2</code>,<code class="varname">y2</code>. <code class="varname">x1</code>,<code
class="varname">y1</code>, <code class="varname">x2</code>,<code class="varname">y2</code> může být pro delší
lom
ené čáry nahrazeno maticí <code class="varname">n</code> krát 2. Připadně vektor <code
class="varname">v</code> může být sloupcový vektor komplexních čísel, což je matice <code
class="varname">n</code> krát 1 a jednotlivá komplexní čísla jsou pak považována za body v
rovině.</p><p>Mohou být přidány dodatečné parametry, které určují barvu, tloušťku a šipky čáry a vykreslení
okna nebo legendy. Stačí přidat argument v podobě řetězce <strong
class="userinput"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>, <strong
class="userinput"><code>"arrow"</code></strong> nebo <strong class="userinput"><code>"legend"</code></strong>
a za ním určit barvu, tloušťku, okno jako 4prvkový vektor, typ šipky nebo legendu. (Šipka a okno jsou
podporovány od verze 1.0.6.)</p><p>Pokud je čára považovaná za vyplněný mnohoúhelník, vyplněný dan
ou barvou, můžete zadat argument <strong class="userinput"><code>"filled"</code></strong>. K dispozici od
verze 1.0.22.</p><p>Barva by měla být buď řetězec symbolizující běžným anglickým slovem barvu, kterou
rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
apod. Nebo druhou možností je zadat barvu ve formátu RGB jako <strong
class="userinput"><code>"#rgb"</code></strong>, <strong class="userinput"><code>"#rrggbb"</code></strong>
nebo <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové
soustavy červené, zelené a modré složky barvy. A nakonec třetí možností, od verze 1.0.18, je také určení
barvy vektorem reálných čísel, která představují červenou, zelenou a modrou složku v rozmezí 0 až 1, např.
<strong class="userinput"><code>[1.0,0.5,0.1]</cod
e></strong>.</p><p>Okno by mělo být zadáno buď obvyklým způsobem jako <strong
class="userinput"><code>[x1,x2,y1,y2]</code></strong> nebo alternativně může být použit řetězec <strong
class="userinput"><code>"fit"</code></strong>, v kterémž to případě bude rozsah x určen přesně a rozsah y
bude nastaven s pětiprocentním přesahem křivky.</p><p>Specifikace šipky by měla být <strong
class="userinput"><code>"origin"</code></strong> (počátek), <strong
class="userinput"><code>"end"</code></strong> (konec), <strong class="userinput"><code>"both"</code></strong>
(obojí) nebo <strong class="userinput"><code>"none"</code></strong> (nic).</p><p>A nakonec legenda, která by
měla být zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě, že se legenda
tiskne.</p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","Řešení")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
-</pre><p>Na rozdíl od většiny ostatních funkcí, u kterých je jedno, jestli je předán sloupcový nebo řádkový
vektor, při zadávání bodů v podobě vektoru komplexních čísel je kvůli možným nejednoznačnostem nutné vždy
zadat sloupcový vektor.</p><p>Zadávání <code class="varname">v</code> jako sloupcového vektoru komplexních
čísel je implementováno od verze 1.0.22.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints
(v,...)</pre><p>Vykreslit bod v <code class="varname">x</code>, <code class="varname">y</code>. Vstupem může
být matice <code class="varname">n</code> krát 2 pro <code class="varname">n</code> různých bodů. Tato funkce
má v podstatě stejné vstupní údaje jako <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>. Případně vektor <code
class="varname">v</code> může být sloupcový vektor komplexních čísel, což je matice <code
class="varname">n</code> krát 1 a jednotlivá komplexní čísla jsou považována za body v rovině.</p><p>Mohou
být přidány dodatečné parametry, které určují barvu a tloušťku čáry a vykreslení okna nebo legendy. Stačí
přidat argument v podobě řetězce <strong class="userinput"><code>"color"</code></strong>, <strong
class="userinput"><code>"thickness"</code></strong>, <strong class="userinput"><code>"window"</code></strong>
nebo <strong class="userinput"><code>"legend"</code></strong> a za ním určit barvu, tloušťku, okno jako
4prvkový vektor nebo legendu.</p><p>Barva by měla být buď řetězec symbolizující běžným anglickým slovem
barvu, kterou rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
apod. Nebo druh
ou možností je zadat barvu ve formátu RGB jako <strong class="userinput"><code>"#rgb"</code></strong>,
<strong class="userinput"><code>"#rrggbb"</code></strong> nebo <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové soustavy
červené, zelené a modré složky barvy. A nakonec třetí možností je také určení barvy vektorem reálných čísel,
která představují červenou, zelenou a modrou složku v rozmezí 0 až 1.</p><p>Okno by mělo být zadáno buď
obvyklým způsobem jako <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong> nebo alternativně může
být použit řetězec <strong class="userinput"><code>"fit"</code></strong>, v kterémž to případě bude rozsah x
určen přesně a rozsah y bude nastaven s pětiprocentním přesahem křivky.</p><p>A nakonec legenda, která by
měla být zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě, že se lege
nda tiskne.</p><p>Examples: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
+</pre><p>
+ </p><p>Na rozdíl od většiny ostatních funkcí, u kterých je jedno, jestli je předán sloupcový nebo
řádkový vektor, při zadávání bodů v podobě vektoru komplexních čísel je kvůli možným nejednoznačnostem nutné
vždy zadat sloupcový vektor.</p><p>Zadávání <code class="varname">v</code> jako sloupcového vektoru
komplexních čísel je implementováno od verze 1.0.22.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints
(v,...)</pre><p>Vykreslit bod v <code class="varname">x</code>, <code class="varname">y</code>. Vstupem může
být matice <code class="varname">n</code> krát 2 pro <code class="varname">n</code> různých bodů. Tato funkce
má v podstatě stejné vstupní údaje jako <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>. Případně vekt
or <code class="varname">v</code> může být sloupcový vektor komplexních čísel, což je matice <code
class="varname">n</code> krát 1 a jednotlivá komplexní čísla jsou považována za body v rovině.</p><p>Mohou
být přidány dodatečné parametry, které určují barvu a tloušťku čáry a vykreslení okna nebo legendy. Stačí
přidat argument v podobě řetězce <strong class="userinput"><code>"color"</code></strong>, <strong
class="userinput"><code>"thickness"</code></strong>, <strong class="userinput"><code>"window"</code></strong>
nebo <strong class="userinput"><code>"legend"</code></strong> a za ním určit barvu, tloušťku, okno jako
4prvkový vektor nebo legendu.</p><p>Barva by měla být buď řetězec symbolizující běžným anglickým slovem
barvu, kterou rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
apod. N
ebo druhou možností je zadat barvu ve formátu RGB jako <strong
class="userinput"><code>"#rgb"</code></strong>, <strong class="userinput"><code>"#rrggbb"</code></strong>
nebo <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové
soustavy červené, zelené a modré složky barvy. A nakonec třetí možností je také určení barvy vektorem
reálných čísel, která představují červenou, zelenou a modrou složku v rozmezí 0 až 1.</p><p>Okno by mělo být
zadáno buď obvyklým způsobem jako <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong> nebo
alternativně může být použit řetězec <strong class="userinput"><code>"fit"</code></strong>, v kterémž to
případě bude rozsah x určen přesně a rozsah y bude nastaven s pětiprocentním přesahem křivky.</p><p>A nakonec
legenda, která by měla být zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě,
že
se legenda tiskne.</p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","Řešení")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","Sedmá
odmocnina z jednotky")</code></strong>
-</pre><p>Na rozdíl od většiny ostatních funkcí, u kterých je jedno, jestli jim předáte sloupcový nebo
řádkový vektor, může u předávání bodu v podobě vektoru komplexních čísel docházet k nejednoznačnostem. Proto
musíte vždy předat sloupcový vektor. Všimněte si v posledním příkladu transpozice vektoru <strong
class="userinput"><code>0:6</code></strong>, aby se z něj stal sloupcový vektor.</p><p>Dostupné od verze
1.0.18. Zadávání <code class="varname">v</code> v podobě sloupcového vektoru komplexních čísel je
implementováno od verze 1.0.22.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotMouseLocation"></a>LinePlotMouseLocation</span></dt><dd><pre
class="synopsis">LinePlotMouseLocation ()</pre><p>Vrátit řádkový vektor v kreslení odpovídající aktuální
pozici myši. Pokud kreslení není viditelné, vypíše se chyba a vrátí <code class="constant">null</code>. V
takovém případě byste měli spustit <a class=
"link" href="ch11s20.html#gel-function-LinePlot"><code class="function">LinePlot</code></a> nebo <a
class="link" href="ch11s20.html#gel-function-LinePlotClear"><code class="function">LinePlotClear</code></a>,
abyste přepnuli okno s grafem do režimu kreslení. Viz také <a class="link"
href="ch11s20.html#gel-function-LinePlotWaitForClick"><code
class="function">LinePlotWaitForClick</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotParametric"></a>LinePlotParametric</span></dt><dd><pre
class="synopsis">LinePlotParametric (xfce,yfce,...)</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust)</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust,[x1,x2,y1,y2])</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust, "fit")</pre><p>Vykreslit parametrickou funkci v podobě čárového grafu. Jako první
se zadávají funk
ce pro <code class="varname">x</code> a <code class="varname">y</code>, následované volitelnými omezeními
<strong class="userinput"><code>t1,t2,tprirust</code></strong> pro <code class="varname">t</code> a pak
mezemi v podobě <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Pokud žádné meze nejsou
zadány, použijí se aktuálně nastavené meze (viz <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Místo mezí
x a y je možné zadat řetězec "fit" a meze se pak zvolí podle maximálního rozsahu grafu.</p><p>Parametr <a
class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> ovládá vykreslování legendy.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotWaitForClick"></a>LinePlotWaitForClick</span></dt><dd><pre
class="synopsis">LinePlotWaitForClick ()</pre><p>Pokud je v režimu kreslení, čeká na kliknut
í v kreslícím okně a následně vrátí pozici kliknutí v podobě řádkového vektoru. Pokud je okno zavřené, vrátí
se funkce okamžitě s hodnotou <code class="constant">null</code>. Pokud okno není v režimu kreslení, přepne
jej do něj a, pokud není zobrazené, zobrazí jej. Viz také <a class="link"
href="ch11s20.html#gel-function-LinePlotMouseLocation"><code
class="function">LinePlotMouseLocation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasFreeze"></a>PlotCanvasFreeze</span></dt><dd><pre
class="synopsis">PlotCanvasFreeze ()</pre><p>Dočasně zmrazí vykreslování grafu na plátno. To se hodí, když
kreslíte spoustu prvků a chcete to pozdržet, aby se fyzicky vykreslilo až všechno naráz a předešlo se tím
blikání. Až máte veškeré kreslení hotovo, měli byste zavolat funkci <a class="link"
href="ch11s20.html#gel-function-PlotCanvasThaw"><code class="function">PlotCanvasThaw</code></a>.</p><p>Na
konci jakéhokoli
v provádění je plátno automaticky rozmrazeno, takže by se nemělo stát, že zůstane zmrazené. Kupříkladu ve
chvíli, kdy se zobrazí nový příkazový řádek, dojde k automatickému rozmrazení. Také si všimněte, že volání
zmrazení a rozmrazení mohou být zanořená.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasThaw"></a>PlotCanvasThaw</span></dt><dd><pre class="synopsis">PlotCanvasThaw
()</pre><p>Rozmrazí plátno pro vykreslování grafu zmrazené pomocí <a class="link"
href="ch11s20.html#gel-function-PlotCanvasFreeze"><code class="function">PlotCanvasFreeze</code></a> a ihned
jej překreslí. Platno je také rozmrazeno vždy po skončení provádění libovolného programu.</p><p>Verze 1.0.18
a novější.</p></dd><dt><span class="term"><a
name="gel-function-PlotWindowPresent"></a>PlotWindowPresent</span></dt><dd><pre
class="synopsis">PlotWindowPresent ()</pre><p>Zobrazí a přenese do popředí vykre
slovací okno, případně jej vytvoří, pokud je třeba. Normálně je okno vytvořeno, když je zavolána některá z
kreslících funkcí, ale nemusí být vždy přeneseno do popředí, když je schováno za jinými okny. Proto je dobré
volat tento kript, když bylo vykreslovací okno vytvořeno již dříve a nyní je schováno za konzolí nebo jinými
okny.</p><p>Verze 1.0.19 a novější.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldClearSolutions"></a>SlopefieldClearSolutions</span></dt><dd><pre
class="synopsis">SlopefieldClearSolutions ()</pre><p>Vymazat řešení vykreslená funkcí <a class="link"
href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldDrawSolution"></a>SlopefieldDrawSolution</span></dt><dd><pre
class="synopsis">SlopefieldDrawSolution (x, y, dx)</pre><p>Když je aktivní vykreslování směrového pole, vykre
slí řešení se zadanou počáteční podmínkou. Použita je standardní Rungeho-Kuttova metoda s přírůstkem <code
class="varname">dx</code>. Řešení v grafu zůstanou, dokud není zobrazen jiný graf nebo není zavolána funkce
<a class="link" href="ch11s20.html#gel-function-SlopefieldClearSolutions"><code
class="function">SlopefieldClearSolutions</code></a>. Pro vykreslení řešení můžete použít i grafické rozhraní
a počáteční podmínky zadat pomocí myši.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldPlot"></a>SlopefieldPlot</span></dt><dd><pre class="synopsis">SlopefieldPlot
(fce)</pre><pre class="synopsis">SlopefieldPlot (fce,x1,x2,y1,y2)</pre><p>Vykreslit směrové pole. Funkce
<code class="varname">fce</code> by měla přebírat dvě reálná čísla <code class="varname">x</code> a <code
class="varname">y</code> nebo jedno komplexní číslo. Volitelně můžete zadat meze vykreslovacího okna jako
souřadnice <code class="varnam
e">x1</code>, <code class="varname">x2</code>, <code class="varname">y1</code>, <code
class="varname">y2</code>. Pokud žádné meze nejsou zadány, použijí se aktuálně nastavení mezí (viz <a
class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).</p><p>Parametr <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
ovládá vykreslování legendy.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
+</pre><p>
+ </p><p>Na rozdíl od většiny ostatních funkcí, u kterých je jedno, jestli jim předáte sloupcový
nebo řádkový vektor, může u předávání bodu v podobě vektoru komplexních čísel docházet k nejednoznačnostem.
Proto musíte vždy předat sloupcový vektor. Všimněte si v posledním příkladu transpozice vektoru <strong
class="userinput"><code>0:6</code></strong>, aby se z něj stal sloupcový vektor.</p><p>Dostupné od verze
1.0.18. Zadávání <code class="varname">v</code> v podobě sloupcového vektoru komplexních čísel je
implementováno od verze 1.0.22.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotMouseLocation"></a>LinePlotMouseLocation</span></dt><dd><pre
class="synopsis">LinePlotMouseLocation ()</pre><p>Vrátit řádkový vektor v kreslení odpovídající aktuální
pozici myši. Pokud kreslení není viditelné, vypíše se chyba a vrátí <code class="constant">null</code>. V
takovém případě byste měli spustit <
a class="link" href="ch11s20.html#gel-function-LinePlot"><code class="function">LinePlot</code></a> nebo <a
class="link" href="ch11s20.html#gel-function-LinePlotClear"><code class="function">LinePlotClear</code></a>,
abyste přepnuli okno s grafem do režimu kreslení. Viz také <a class="link"
href="ch11s20.html#gel-function-LinePlotWaitForClick"><code
class="function">LinePlotWaitForClick</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotParametric"></a>LinePlotParametric</span></dt><dd><pre
class="synopsis">LinePlotParametric (xfce,yfce,...)</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust)</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust,[x1,x2,y1,y2])</pre><pre class="synopsis">LinePlotParametric
(xfce,yfce,t1,t2,tprirust, "fit")</pre><p>Vykreslit parametrickou funkci v podobě čárového grafu. Jako první
se zadáva
jí funkce pro <code class="varname">x</code> a <code class="varname">y</code>, následované volitelnými
omezeními <strong class="userinput"><code>t1,t2,tprirust</code></strong> pro <code class="varname">t</code> a
pak mezemi v podobě <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Pokud žádné meze
nejsou zadány, použijí se aktuálně nastavené meze (viz <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Místo mezí
x a y je možné zadat řetězec "fit" a meze se pak zvolí podle maximálního rozsahu grafu.</p><p>Parametr <a
class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> ovládá vykreslování legendy.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotWaitForClick"></a>LinePlotWaitForClick</span></dt><dd><pre
class="synopsis">LinePlotWaitForClick ()</pre><p>Pokud je v režimu kreslení, čeká na
kliknutí v kreslícím okně a následně vrátí pozici kliknutí v podobě řádkového vektoru. Pokud je okno
zavřené, vrátí se funkce okamžitě s hodnotou <code class="constant">null</code>. Pokud okno není v režimu
kreslení, přepne jej do něj a, pokud není zobrazené, zobrazí jej. Viz také <a class="link"
href="ch11s20.html#gel-function-LinePlotMouseLocation"><code
class="function">LinePlotMouseLocation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasFreeze"></a>PlotCanvasFreeze</span></dt><dd><pre
class="synopsis">PlotCanvasFreeze ()</pre><p>Dočasně zmrazí vykreslování grafu na plátno. To se hodí, když
kreslíte spoustu prvků a chcete to pozdržet, aby se fyzicky vykreslilo až všechno naráz a předešlo se tím
blikání. Až máte veškeré kreslení hotovo, měli byste zavolat funkci <a class="link"
href="ch11s20.html#gel-function-PlotCanvasThaw"><code class="function">PlotCanvasThaw</code></a>.</p><p>Na
konci jak
éhokoliv provádění je plátno automaticky rozmrazeno, takže by se nemělo stát, že zůstane zmrazené.
Kupříkladu ve chvíli, kdy se zobrazí nový příkazový řádek, dojde k automatickému rozmrazení. Také si
všimněte, že volání zmrazení a rozmrazení mohou být zanořená.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span
class="term"><a name="gel-function-PlotCanvasThaw"></a>PlotCanvasThaw</span></dt><dd><pre
class="synopsis">PlotCanvasThaw ()</pre><p>Rozmrazí plátno pro vykreslování grafu zmrazené pomocí <a
class="link" href="ch11s20.html#gel-function-PlotCanvasFreeze"><code
class="function">PlotCanvasFreeze</code></a> a ihned jej překreslí. Platno je také rozmrazeno vždy po
skončení provádění libovolného programu.</p><p>Verze 1.0.18 a novější.</p></dd><dt><span class="term"><a
name="gel-function-PlotWindowPresent"></a>PlotWindowPresent</span></dt><dd><pre
class="synopsis">PlotWindowPresent ()</pre><p>Zobrazí a přenese do popřed
í vykreslovací okno, případně jej vytvoří, pokud je třeba. Normálně je okno vytvořeno, když je zavolána
některá z kreslících funkcí, ale nemusí být vždy přeneseno do popředí, když je schováno za jinými okny. Proto
je dobré volat tento kript, když bylo vykreslovací okno vytvořeno již dříve a nyní je schováno za konzolí
nebo jinými okny.</p><p>Verze 1.0.19 a novější.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldClearSolutions"></a>SlopefieldClearSolutions</span></dt><dd><pre
class="synopsis">SlopefieldClearSolutions ()</pre><p>Vymazat řešení vykreslená funkcí <a class="link"
href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldDrawSolution"></a>SlopefieldDrawSolution</span></dt><dd><pre
class="synopsis">SlopefieldDrawSolution (x, y, dx)</pre><p>Když je aktivní vykreslování směrového pol
e, vykreslí řešení se zadanou počáteční podmínkou. Použita je standardní Rungeho-Kuttova metoda s přírůstkem
<code class="varname">dx</code>. Řešení v grafu zůstanou, dokud není zobrazen jiný graf nebo není zavolána
funkce <a class="link" href="ch11s20.html#gel-function-SlopefieldClearSolutions"><code
class="function">SlopefieldClearSolutions</code></a>. Pro vykreslení řešení můžete použít i grafické rozhraní
a počáteční podmínky zadat pomocí myši.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldPlot"></a>SlopefieldPlot</span></dt><dd><pre class="synopsis">SlopefieldPlot
(fce)</pre><pre class="synopsis">SlopefieldPlot (fce,x1,x2,y1,y2)</pre><p>Vykreslit směrové pole. Funkce
<code class="varname">fce</code> by měla přebírat dvě reálná čísla <code class="varname">x</code> a <code
class="varname">y</code> nebo jedno komplexní číslo. Volitelně můžete zadat meze vykreslovacího okna jako
souřadnice <code class
="varname">x1</code>, <code class="varname">x2</code>, <code class="varname">y1</code>, <code
class="varname">y2</code>. Pokud žádné meze nejsou zadány, použijí se aktuálně nastavení mezí (viz <a
class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).</p><p>Parametr <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
ovládá vykreslování legendy.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</code></strong>
</pre></dd><dt><span class="term"><a name="gel-function-SurfacePlot"></a>SurfacePlot</span></dt><dd><pre
class="synopsis">SurfacePlot (fce)</pre><pre class="synopsis">SurfacePlot (fce,x1,x2,y1,y2,z1,z2)</pre><pre
class="synopsis">SurfacePlot (fce,x1,x2,y1,y2)</pre><pre class="synopsis">SurfacePlot
(fce,[x1,x2,y1,y2,z1,z2])</pre><pre class="synopsis">SurfacePlot (fce,[x1,x2,y1,y2])</pre><p>Vykreslit funkci
plochy, která přebírá buď dva argumenty nebo komplexní číslo. Jako první se předává funkce, pak následují
meze <code class="varname">x1</code>, <code class="varname">x2</code>, <code class="varname">y1</code>, <code
class="varname">y2</code>, <code class="varname">z1</code>, <code class="varname">z2</code>. Pokud žádné meze
nejsou zadány, použijí se aktuálně nastavené meze (viz <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">SurfacePlotWindow</code></a>). V
současnosti umí Genius vykreslovat jen funkci jedné p
lochy.</p><p>Když nejsou meze zadány, použije se pro ně minimum a maximum funkce.</p><p>Příklady: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(|sin|,-1,1,-1,1,0,1.5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(`(x,y)=x^2+y,-1,1,-1,1,-2,2)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(`(z)=|z|^2,-1,1,-1,1,0,2)</code></strong>
@@ -27,7 +33,7 @@
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(data,[-1,1,-1,1],"Moje data")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>d:=null; for i=1 to 20 do for j=1 to
10 do d@(i,j) = (0.1*i-1)^2-(0.1*j)^2;</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(d,[-1,1,0,1],"poloviční sedlo")</code></strong>
-</pre><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>Vykreslit čáru z <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> do <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code>. <code class="varname">x1</code>, <code class="varname">y1</code>, <code
class="varname">z1</code>, <code class="varname">x2</code>, <code class="varname">y2</code>, <code
class="varname">z2</code> může být pro delší lomené čáry nahrazeno maticí <code class="varname">n</code> krát
3.</p><p>Mohou být přidány dodatečné parametry, které určují barvu a tloušťku čáry, šipky a vykreslení okna
nebo legendy. Stačí přidat argument v podobě řetězce <strong class="userinput
"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>, <strong
class="userinput"><code>"window"</code></strong> nebo <strong
class="userinput"><code>"legend"</code></strong> a za ním určit barvu, tloušťku, okno jako 6prvkový vektor
nebo legendu.</p><p>Barva by měla být buď řetězec symbolizující běžným anglickým slovem barvu, kterou
rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
apod. Nebo druhou možností je zadat barvu ve formátu RGB jako <strong
class="userinput"><code>"#rgb"</code></strong>, <strong class="userinput"><code>"#rrggbb"</code></strong>
nebo <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové
soustavy červené, zelené a modré složky barvy. A nakonec třetí možností, od verze 1.0.18, je také určení b
arvy vektorem reálných čísel, která představují červenou, zelenou a modrou složku v rozmezí 0 až 1, např.
<strong class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Okno by mělo být zadáno buď obvyklým
způsobem jako <strong class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong> nebo alternativně může být
použit řetězec <strong class="userinput"><code>"fit"</code></strong>, v kterémž to případě bude rozsah x
určen přesně a rozsah y bude nastaven s pětiprocentním přesahem křivky.</p><p>A nakonec legenda, která by
měla být zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě, že se legenda
tiskne.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
+</pre><p>Verze 1.0.16 a novější.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>Vykreslit čáru z <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> do <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code>. <code class="varname">x1</code>, <code class="varname">y1</code>, <code
class="varname">z1</code>, <code class="varname">x2</code>, <code class="varname">y2</code>, <code
class="varname">z2</code> může být pro delší lomené čáry nahrazeno maticí <code class="varname">n</code> krát
3.</p><p>Mohou být přidány dodatečné parametry, které určují barvu a tloušťku čáry a vykreslení okna nebo
legendy. Stačí přidat argument v podobě řetězce <strong class="userinput"><code>
"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>, <strong
class="userinput"><code>"window"</code></strong> nebo <strong
class="userinput"><code>"legend"</code></strong> a za ním určit barvu, tloušťku, okno jako 6prvkový vektor
nebo legendu.</p><p>Barva by měla být buď řetězec symbolizující běžným anglickým slovem barvu, kterou
rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>, <strong
class="userinput"><code>"blue"</code></strong>, <strong class="userinput"><code>"yellow"</code></strong>,
apod. Nebo druhou možností je zadat barvu ve formátu RGB jako <strong
class="userinput"><code>"#rgb"</code></strong>, <strong class="userinput"><code>"#rrggbb"</code></strong>
nebo <strong class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové
soustavy červené, zelené a modré složky barvy. A nakonec třetí možností, od verze 1.0.18, je také určení
barvy vek
torem reálných čísel, která představují červenou, zelenou a modrou složku v rozmezí 0 až 1, např. <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Okno by mělo být zadáno buď obvyklým způsobem
jako <strong class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong> nebo alternativně může být použit
řetězec <strong class="userinput"><code>"fit"</code></strong>, v kterémž to případě bude rozsah x určen
přesně a rozsah y bude nastaven s pětiprocentním přesahem křivky.</p><p>A nakonec legenda, která by měla být
zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě, že se legenda
tiskne.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine([0,0,0;1,-1,2;-1,-1,-3])</code></strong>
</pre><p>Dostupné ve verzi 1.0.19 a novějších.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawPoints"></a>SurfacePlotDrawPoints</span></dt><dd><pre
class="synopsis">SurfacePlotDrawPoints (x,y,z,...)</pre><pre class="synopsis">SurfacePlotDrawPoints
(v,...)</pre><p>Vykreslit bod v <code class="varname">x</code>,<code class="varname">y</code>,<code
class="varname">z</code>. Vstupem může být matice <code class="varname">n</code> krát 3 pro <code
class="varname">n</code> různých bodů. Tato funkce má v podstatě stejné vstupní údaje jako <a class="link"
href="ch11s20.html#gel-function-SurfacePlotDrawLine">SurfacePlotDrawLine</a>.</p><p>Mohou být přidány
dodatečné parametry, které určují barvu a tloušťku čáry a vykreslení okna nebo legendy. Stačí přidat argument
v podobě řetězce <strong class="userinput"><code>"color"</code></strong>, <strong
class="userinput"><code>"thickness"</code></strong>, <strong class="userinput"><code>"
window"</code></strong> nebo <strong class="userinput"><code>"legend"</code></strong> a za ním určit barvu,
tloušťku, okno jako 6prvkový vektor nebo legendu.</p><p>Barva by měla být buď řetězec symbolizující běžným
anglickým slovem barvu, kterou rozpozná GTK, jako <strong class="userinput"><code>"red"</code></strong>,
<strong class="userinput"><code>"blue"</code></strong>, <strong
class="userinput"><code>"yellow"</code></strong>, apod. Nebo druhou možností je zadat barvu ve formátu RGB
jako <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> nebo <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, kde r, g a b jsou číslice šestnáctkové soustavy
červené, zelené a modré složky barvy. A nakonec třetí možností je také určení barvy vektorem reálných čísel,
která představují červenou, zelenou a modrou složku v rozmezí 0 až 1.</p><p>Okno by mělo být zadáno b
uď obvyklým způsobem jako <strong class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong> nebo
alternativně může být použit řetězec <strong class="userinput"><code>"fit"</code></strong>, v kterémž to
případě bude rozsah x určen přesně a rozsah y bude nastaven s pětiprocentním přesahem křivky.</p><p>A nakonec
legenda, která by měla být zadána jako řetězec, který se použije k osvětlení grafu. Samozřejmě jen v případě,
že se legenda tiskne.</p><p>Příklady: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints([0,0,0;1,-1,2;-1,-1,1])</code></strong>
diff --git a/help/cs/html/index.html b/help/cs/html/index.html
index 97827e1..f3b4f05 100644
--- a/help/cs/html/index.html
+++ b/help/cs/html/index.html
@@ -1,3 +1,3 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Příručka k aplikaci
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Příručka k matematickému nástroji Genius."><link rel="home" href="index.html" title="Příručka k
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width="20%" align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a
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class="titlepage"><div><div><h1 class="title"><a name="index"></a>Příručka k aplikaci
Genius</h1></div><div><div class="authorgroup">
<div class="author"><h3 class="author"><span class="firstname">Jiří</span> <span
class="surname">Lebl</span></h3><div class="affiliation"><span class="orgname">Státní oklahamská
univerzita<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code> </p></div></div></div><div class="author"><h3 class="author"><span
class="firstname">Kai</span> <span class="surname">Willadsen</span></h3><div class="affiliation"><span
class="orgname">Univerzita Queensland, Austrálie<br></span><div class="address"><p> <code
class="email"><<a class="email" href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">Tato příručka popisuje aplikaci Genius ve verzi
1.0.22.</p></div><div><p class="copyright">Copyright © 1997 – 2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><div c
lass="legalnotice"><a name="legalnotice"></a><p>Je povoleno kopírovat, šířit a/nebo upravovat tento dokument
za podmínek GNU Free Documentation License (GFDL) ve verzi 1.1 nebo v jakékoli další verzi vydané nadací Free
Software Foundation; bez neměnných oddílů, bez textů předních desek a bez textů zadních desek. Kopii licence
GFDL naleznete pod <a class="ulink" href="ghelp:fdl" target="_top">tímto odkazem</a> nebo v souboru
COPYING-DOCS dodávaném s touto příručkou.</p><p>Tato příručka je součástí sbírky příruček GNOME šířených za
podmínek licence GFDL. Pokud chcete tento dokument šířit odděleně od sbírky, musíte přiložit kopii licence
dle popisu v oddílu 6 dané licence.</p><p>Mnoho názvů použitých firmami k zviditelnění produktů nebo služeb
jsou ochranné známky. Na místech, kde jsou tyto názvy v dokumentaci použity a členové Dokumentačního projektu
GNOME jsou si vědomi skutečnosti, že se jedná o ochranno
u známku, je takovýto název psán velkými písmeny celý nebo s velkým písmenem na začátku.</p><p>DOKUMENT A
JEHO UPRAVENÉ VERZE JSOU ŠÍŘENY V SOULADU SE ZNĚNÍM LICENCE GNU FREE DOCUMENTATION LICENSE S NÁSLEDUJÍCÍM
USTANOVENÍM: </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>DOKUMENT
JE POSKYTOVÁN V PODOBĚ „JAK JE“, BEZ ZÁRUKY JAKÉHOKOLIV DRUHU, NEPOSKYTUJÍ SE ANI ODVOZENÉ ZÁRUKY, ZÁRUKY, ŽE
DOKUMENT, NEBO JEHO UPRAVENÁ VERZE, JE BEZCHYBNÝ, NEBO ZÁRUKY PRODEJNOSTI, VHODNOSTI PRO URČITÝ ÚČEL, NEBO
NEPORUŠENOSTI. RIZIKO NEKVALITY, NEPŘESNOSTI A ŠPATNÉHO PROVEDENÍ DOKUMENTU, NEBO JEHO UPRAVENÉ VERZE, NESETE
VY. POKUD JE TENTO DOKUMENT NEBO JEHO UPRAVENÁ VERZE VADNÁ V JAKÉMKOLIV SMYSLU, VY (NIKOLIV PŮVODCE, AUTOR
NEBO JAKÝKOLIV PŘISPĚVATEL) PŘEBÍRÁTE ODPOVĚDNOST ZA JAKÉKOLIV NÁKLADY NA NUTNÉ ÚPRAVY, OPRAVY ČI SLUŽBY.
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�TO LICENCE. BEZ TOHOTO PROHLÁŠENÍ NENÍ PODLE TÉTO DOHODY POVOLENO UŽÍVÁNÍ ANI ÚPRAVY TOHOTO DOKUMENTU;
DÁLE</p></li><li class="listitem"><p>ZA ŽÁDNÝCH OKOLNOSTÍ A ŽÁDNÝCH PRÁVNÍCH PŘEDPOKLADŮ, AŤ SE JEDNÁ O
PŘEČIN (VČETNĚ NEDBALOSTNÍCH), SMLOUVU NEBO JINÉ, NENÍ AUTOR, PŮVODNÍ PISATEL, KTERÝKOLIV PŘISPĚVATEL NEBO
KTERÝKOLIV DISTRIBUTOR TOHOTO DOKUMENTU NEBO UPRAVENÉ VERZE DOKUMENTU NEBO KTERÝKOLIV DODAVATEL NĚKTERÉ Z
TĚCHTO STRAN ODPOVĚDNÝ NĚJAKÉ OSOBĚ ZA PŘÍMÉ, NEPŘÍMÉ, SPECIÁLNÍ, NAHODILÉ NEBO NÁSLEDNÉ ŠKODY JAKÉHOKOLIV
CHARAKTERU, VČETNĚ, ALE NEJEN, ZA POŠKOZENÍ ZE ZTRÁTY DOBRÉHO JMÉNA, PŘERUŠENÍ PRÁCE, PORUCHY NEBO NESPRÁVNÉ
FUNKCE POČÍTAČE NEBO JINÉHO A VŠECH DALŠÍCH ŠKOD NEBO ZTRÁT VYVSTÁVAJÍCÍCH Z NEBO VZTAHUJÍCÍCH SE K POUŽÍVÁNÍ
TOHOTO DOKUMENTU NEBO UPRAVENÝCH VERZÍ DOKUMENTU, I KDYŽ BY TAKOVÁTO STRANA BYLA INFORMOVANÁ O MOŽNOSTI
TAKOVÉHOTO POŠKOZENÍ.</p></li></ol></d
iv></div></div><div><div class="legalnotice"><a name="idm45682164828672"></a><p
class="legalnotice-title"><b>Ohlasy</b></p><p>Pokud chcete oznámit chybu nebo navrhnout vylepšení vztahující
se k aplikaci <span class="application">matematický nástroj Genius</span> nebo této příručce, navštivte
prosím <a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">webovou stránku aplikace
Genius</a> nebo napište autorovi na e-mail <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code>.</p></div></div><div><div class="revhistory"><table
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class="titlepage"><div><div><h1 class="title"><a name="index"></a>Příručka k aplikaci
Genius</h1></div><div><div class="authorgroup">
<div class="author"><h3 class="author"><span class="firstname">Jiří</span> <span
class="surname">Lebl</span></h3><div class="affiliation"><span class="orgname">Státní oklahamská
univerzita<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code> </p></div></div></div><div class="author"><h3 class="author"><span
class="firstname">Kai</span> <span class="surname">Willadsen</span></h3><div class="affiliation"><span
class="orgname">Univerzita Queensland, Austrálie<br></span><div class="address"><p> <code
class="email"><<a class="email" href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">Tato příručka popisuje aplikaci Genius ve verzi
1.0.22.</p></div><div><p class="copyright">Copyright © 1997 – 2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><div c
lass="legalnotice"><a name="legalnotice"></a><p>Je povoleno kopírovat, šířit a/nebo upravovat tento dokument
za podmínek GNU Free Documentation License (GFDL) ve verzi 1.1 nebo v jakékoli další verzi vydané nadací Free
Software Foundation; bez neměnných oddílů, bez textů předních desek a bez textů zadních desek. Kopii licence
GFDL naleznete pod <a class="ulink" href="ghelp:fdl" target="_top">tímto odkazem</a> nebo v souboru
COPYING-DOCS dodávaném s touto příručkou.</p><p>Tato příručka je součástí sbírky příruček GNOME šířených za
podmínek licence GFDL. Pokud chcete tento dokument šířit odděleně od sbírky, musíte přiložit kopii licence
dle popisu v oddílu 6 dané licence.</p><p>Mnoho názvů použitých firmami k zviditelnění produktů nebo služeb
jsou ochranné známky. Na místech, kde jsou tyto názvy v dokumentaci použity a členové Dokumentačního projektu
GNOME jsou si vědomi skutečnosti, že se jedná o ochranno
u známku, je takovýto název psán velkými písmeny celý nebo s velkým písmenem na začátku.</p><p>DOKUMENT A
JEHO UPRAVENÉ VERZE JSOU ŠÍŘENY V SOULADU SE ZNĚNÍM LICENCE GNU FREE DOCUMENTATION LICENSE S NÁSLEDUJÍCÍM
USTANOVENÍM: </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>DOKUMENT
JE POSKYTOVÁN V PODOBĚ „JAK JE“, BEZ ZÁRUKY JAKÉHOKOLIV DRUHU, NEPOSKYTUJÍ SE ANI ODVOZENÉ ZÁRUKY, ZÁRUKY, ŽE
DOKUMENT, NEBO JEHO UPRAVENÁ VERZE, JE BEZCHYBNÝ, NEBO ZÁRUKY PRODEJNOSTI, VHODNOSTI PRO URČITÝ ÚČEL, NEBO
NEPORUŠENOSTI. RIZIKO NEKVALITY, NEPŘESNOSTI A ŠPATNÉHO PROVEDENÍ DOKUMENTU, NEBO JEHO UPRAVENÉ VERZE, NESETE
VY. POKUD JE TENTO DOKUMENT NEBO JEHO UPRAVENÁ VERZE VADNÁ V JAKÉMKOLIV SMYSLU, VY (NIKOLIV PŮVODCE, AUTOR
NEBO JAKÝKOLIV PŘISPĚVATEL) PŘEBÍRÁTE ODPOVĚDNOST ZA JAKÉKOLIV NÁKLADY NA NUTNÉ ÚPRAVY, OPRAVY ČI SLUŽBY.
TOTO PROHLÁŠENÍ O ZÁRUCE PŘEDSTAVUJE ZÁKLADNÍ SOUČÁST T�
�TO LICENCE. BEZ TOHOTO PROHLÁŠENÍ NENÍ PODLE TÉTO DOHODY POVOLENO UŽÍVÁNÍ ANI ÚPRAVY TOHOTO DOKUMENTU;
DÁLE</p></li><li class="listitem"><p>ZA ŽÁDNÝCH OKOLNOSTÍ A ŽÁDNÝCH PRÁVNÍCH PŘEDPOKLADŮ, AŤ SE JEDNÁ O
PŘEČIN (VČETNĚ NEDBALOSTNÍCH), SMLOUVU NEBO JINÉ, NENÍ AUTOR, PŮVODNÍ PISATEL, KTERÝKOLIV PŘISPĚVATEL NEBO
KTERÝKOLIV DISTRIBUTOR TOHOTO DOKUMENTU NEBO UPRAVENÉ VERZE DOKUMENTU NEBO KTERÝKOLIV DODAVATEL NĚKTERÉ Z
TĚCHTO STRAN ODPOVĚDNÝ NĚJAKÉ OSOBĚ ZA PŘÍMÉ, NEPŘÍMÉ, SPECIÁLNÍ, NAHODILÉ NEBO NÁSLEDNÉ ŠKODY JAKÉHOKOLIV
CHARAKTERU, VČETNĚ, ALE NEJEN, ZA POŠKOZENÍ ZE ZTRÁTY DOBRÉHO JMÉNA, PŘERUŠENÍ PRÁCE, PORUCHY NEBO NESPRÁVNÉ
FUNKCE POČÍTAČE NEBO JINÉHO A VŠECH DALŠÍCH ŠKOD NEBO ZTRÁT VYVSTÁVAJÍCÍCH Z NEBO VZTAHUJÍCÍCH SE K POUŽÍVÁNÍ
TOHOTO DOKUMENTU NEBO UPRAVENÝCH VERZÍ DOKUMENTU, I KDYŽ BY TAKOVÁTO STRANA BYLA INFORMOVANÁ O MOŽNOSTI
TAKOVÉHOTO POŠKOZENÍ.</p></li></ol></d
iv></div></div><div><div class="legalnotice"><a name="idm48"></a><p
class="legalnotice-title"><b>Ohlasy</b></p><p>Pokud chcete oznámit chybu nebo navrhnout vylepšení vztahující
se k aplikaci <span class="application">matematický nástroj Genius</span> nebo této příručce, navštivte
prosím <a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">webovou stránku aplikace
Genius</a> nebo napište autorovi na e-mail <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code>.</p></div></div><div><div class="revhistory"><table
style="border-style:solid; width:100%;" summary="Přehled revizí"><tr><th align="left" valign="top"
colspan="2"><b>Přehled revizí</b></th></tr><tr><td align="left">Revize 0.2</td><td align="left">Září
2016</td></tr><tr><td align="left" colspan="2">
<p class="author">Jiri (George) Lebl <code class="email"><<a class="email"
href="mailto:jirka 5z com">jirka 5z com</a>></code></p>
</td></tr></table></div></div><div><div class="abstract"><p
class="title"><b>Abstrakt</b></p><p>Příručka k matematickému nástroji
Genius.</p></div></div></div><hr></div><div class="toc"><p><b>Obsah</b></p><dl class="toc"><dt><span
class="chapter"><a href="ch01.html">1. Úvod</a></span></dt><dt><span class="chapter"><a href="ch02.html">2.
Začínáme</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch02.html#genius-to-start">Jak spustit <span
class="application">matematický nástroj Genius</span></a></span></dt><dt><span class="sect1"><a
href="ch02s02.html">Když spustíte aplikaci Genius</a></span></dt></dl></dd><dt><span class="chapter"><a
href="ch03.html">3. Základy používání</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch03.html#genius-usage-workarea">Používání pracovní oblasti</a></span></dt><dt><span class="sect1"><a
href="ch03s02.html">Jak vytvořit nový program</a></span></dt><dt><span class="sect1"><a
href="ch03s03.html">Jak otev�
�ít a spustit program</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch04.html">4.
Vykreslování</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch04.html#genius-line-plots">Čárové
grafy</a></span></dt><dt><span class="sect1"><a href="ch04s02.html">Parametrické
grafy</a></span></dt><dt><span class="sect1"><a href="ch04s03.html">Grafy směrových
polí</a></span></dt><dt><span class="sect1"><a href="ch04s04.html">Grafy vektorových
polí</a></span></dt><dt><span class="sect1"><a href="ch04s05.html">Plošné
grafy</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch05.html">5. Základy jazyka
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch05.html#genius-gel-values">Hodnoty</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-numbers">Čísla</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-booleans">Pravdivostní hodnoty</a></span></dt><dt><span class="sect2"><a
href="
ch05.html#genius-gel-values-strings">Řetězce</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-null">Null</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s02.html">Používání proměnných</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-setting">Nastavování proměnných</a></span></dt><dt><span
class="sect2"><a href="ch05s02.html#genius-gel-variables-built-in">Vestavěné
proměnné</a></span></dt><dt><span class="sect2"><a href="ch05s02.html#genius-gel-previous-result">Proměnná s
posledním výsledkem</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch05s03.html">Používání
funkcí</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-defining">Definování funkcí</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-variable-argument-lists">Proměnný seznam
argumentů</a></span></dt><dt><span class="sect2"><
a href="ch05s03.html#genius-gel-functions-passing-functions">Předávání funkcí
funkcím</a></span></dt><dt><span class="sect2"><a href="ch05s03.html#genius-gel-functions-operations">Operace
s funkcemi</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s04.html">Oddělovač</a></span></dt><dt><span class="sect1"><a
href="ch05s05.html">Komentáře</a></span></dt><dt><span class="sect1"><a href="ch05s06.html">Modulární
aritmetika</a></span></dt><dt><span class="sect1"><a href="ch05s07.html">Seznam operátorů
GEL</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch06.html">6. Programování s jazykem
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch06.html#genius-gel-conditionals">Podmínky</a></span></dt><dt><span class="sect1"><a
href="ch06s02.html">Smyčky</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-while">Smyčky while</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loo
ps-for">Smyčky for</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-foreach">Smyčky foreach</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-break-continue">Break a continue</a></span></dt></dl></dd><dt><span
class="sect1"><a href="ch06s03.html">Součty a součiny</a></span></dt><dt><span class="sect1"><a
href="ch06s04.html">Porovnávací operátory</a></span></dt><dt><span class="sect1"><a
href="ch06s05.html">Globální proměnné a působnost proměnných</a></span></dt><dt><span class="sect1"><a
href="ch06s06.html">Proměnné parametrů</a></span></dt><dt><span class="sect1"><a href="ch06s07.html">Návrat
hodnot</a></span></dt><dt><span class="sect1"><a href="ch06s08.html">Reference</a></span></dt><dt><span
class="sect1"><a href="ch06s09.html">L-hodnoty</a></span></dt></dl></dd><dt><span class="chapter"><a
href="ch07.html">7. Pokročilé programování v jazyce GEL</a></span></dt><dd><dl><dt><span class="sect1"><
a href="ch07.html#genius-gel-error-handling">Obsluha chyb</a></span></dt><dt><span class="sect1"><a
href="ch07s02.html">Syntaxe v nejvyšší úrovni</a></span></dt><dt><span class="sect1"><a
href="ch07s03.html">Vracení funkcí</a></span></dt><dt><span class="sect1"><a href="ch07s04.html">Skutečně
lokální proměnné</a></span></dt><dt><span class="sect1"><a href="ch07s05.html">Spouštěcí procedura
GEL</a></span></dt><dt><span class="sect1"><a href="ch07s06.html">Načítání
programů</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch08.html">8. Matice v jazyce
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch08.html#genius-gel-matrix-support">Zadávání
matic</a></span></dt><dt><span class="sect1"><a href="ch08s02.html">Operátor konjugované transpozice a
transpozice</a></span></dt><dt><span class="sect1"><a href="ch08s03.html">Lineární
algebra</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch09.html">9. Polynomy v jazyce GEL
</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch09.html#genius-gel-polynomials-using">Používání
polynomů</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch10.html">10. Teorie množin v jazyce
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch10.html#genius-gel-sets-using">Používání
množin</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch11.html">11. Seznam funkcí
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch11.html#genius-gel-function-list-commands">Příkazy</a></span></dt><dt><span class="sect1"><a
href="ch11s02.html">Základy</a></span></dt><dt><span class="sect1"><a
href="ch11s03.html">Parametry</a></span></dt><dt><span class="sect1"><a
href="ch11s04.html">Konstanty</a></span></dt><dt><span class="sect1"><a href="ch11s05.html">Práce s
čísly</a></span></dt><dt><span class="sect1"><a href="ch11s06.html">Trigonometrie</a></span></dt><dt><span
class="sect1"><a href="ch11s07.html">Teorie čísel</a
</span></dt><dt><span class="sect1"><a href="ch11s08.html">Práce s maticemi</a></span></dt><dt><span
class="sect1"><a href="ch11s09.html">Lineární algebra</a></span></dt><dt><span class="sect1"><a
href="ch11s10.html">Kombinatorika</a></span></dt><dt><span class="sect1"><a
href="ch11s11.html">Diferenciální/integrální počet </a></span></dt><dt><span class="sect1"><a
href="ch11s12.html">Funkce</a></span></dt><dt><span class="sect1"><a href="ch11s13.html">Řešení
rovnic</a></span></dt><dt><span class="sect1"><a href="ch11s14.html">Statistika</a></span></dt><dt><span
class="sect1"><a href="ch11s15.html">Polynomy</a></span></dt><dt><span class="sect1"><a
href="ch11s16.html">Teorie množin</a></span></dt><dt><span class="sect1"><a href="ch11s17.html">Komutativní
algebra</a></span></dt><dt><span class="sect1"><a href="ch11s18.html">Různé</a></span></dt><dt><span
class="sect1"><a href="ch11s19.html">Symbolické operace</a></span></dt><dt><span class="sect1"><a href="ch1
1s20.html">Vykreslování</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch12.html">12. Příklad
programů v jazyce GEL</a></span></dt><dt><span class="chapter"><a href="ch13.html">13.
Nastavení</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch13.html#genius-prefs-output">Výstup</a></span></dt><dt><span class="sect1"><a
href="ch13s02.html">Přesnost</a></span></dt><dt><span class="sect1"><a
href="ch13s03.html">Terminál</a></span></dt><dt><span class="sect1"><a
href="ch13s04.html">Paměť</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch14.html">14. O <span
class="application">Matematickém nástroji Genius</span></a></span></dt></dl></div><div
class="list-of-figures"><p><b>Seznam obrázků</b></p><dl><dt>2.1. <a href="ch02s02.html#mainwindow-fig">Okno
<span class="application">Matematického nástroje Genius</span></a></dt><dt>4.1. <a
href="ch04.html#lineplot-fig">Okno Vytváření grafu</a></dt><dt>4.2. <a href="ch04.html#lineplot2-fig
">Okno s grafem</a></dt><dt>4.3. <a href="ch04s02.html#paramplot-fig">Karta parametrických
grafů</a></dt><dt>4.4. <a href="ch04s02.html#paramplot2-fig">Parametrické grafy</a></dt><dt>4.5. <a
href="ch04s05.html#surfaceplot-fig">Plošný graf</a></dt></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"> </td><td width="20%"
align="center"> </td><td width="40%" align="right"> <a accesskey="n"
href="ch01.html">Další</a></td></tr><tr><td width="40%" align="left" valign="top"> </td><td width="20%"
align="center"> </td><td width="40%" align="right" valign="top"> Kapitola 1.
Úvod</td></tr></table></div></body></html>
diff --git a/help/de/html/ch05s07.html b/help/de/html/ch05s07.html
index 6507ff9..50ccf37 100644
--- a/help/de/html/ch05s07.html
+++ b/help/de/html/ch05s07.html
@@ -63,10 +63,12 @@ returns 3.
Element by element back division.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>
The mod operator. This does not turn on the <a class="link" href="ch05s06.html" title="Modulare
Auswertung">modular mode</a>, but
- just returns the remainder of <strong class="userinput"><code>a/b</code></strong>.
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
- Element by element the mod operator. Returns the remainder
- after element by element integer <strong class="userinput"><code>a./b</code></strong>.
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
</p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>
Modular evaluation operator. The expression <code class="varname">a</code>
is evaluated modulo <code class="varname">b</code>. See <a class="xref" href="ch05s06.html"
title="Modulare Auswertung">„Modulare Auswertung“</a>.
@@ -102,21 +104,21 @@ returns 3.
greater than or equal to
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
- (can also be combine with the greater than operator).
+ (and they can also be combined with the greater than operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
Less than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
less than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
- (can also be combine with the less than or equal to operator).
+ (they can also be combined with the less than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
Greater than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
greater than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
- (can also be combine with the greater than or equal to operator).
+ (they can also be combined with the greater than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>
Comparison operator. If <code class="varname">a</code> is equal to
<code class="varname">b</code> it returns 0, if <code class="varname">a</code> is less
@@ -136,12 +138,12 @@ returns 3.
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
Logical xor.
- Returns true exactly one of
+ Returns true if exactly one of
<code class="varname">a</code> or <code class="varname">b</code> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
- Logical not. Returns the logical negation of <code class="varname">a</code>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>
@@ -160,7 +162,7 @@ returns 3.
Get element of a matrix in row <code class="varname">b</code> and column
<code class="varname">c</code>. If <code class="varname">b</code>,
<code class="varname">c</code> are vectors, then this gets the corresponding
- rows columns or submatrices.
+ rows, columns or submatrices.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>
Get row of a matrix (or multiple rows if <code class="varname">b</code> is a vector).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Gleiches wie oben.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>
@@ -203,8 +205,8 @@ returns 3.
point numbers and is ever so slightly more precise than
<strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
- Make a imaginary number (multiply <code class="varname">a</code> by the
- imaginary). Note that normally the number <code class="varname">i</code> is
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
</p><pre class="programlisting">(a)*1i
</pre><p>
diff --git a/help/de/html/ch06s05.html b/help/de/html/ch06s05.html
index 473d293..dee7485 100644
--- a/help/de/html/ch06s05.html
+++ b/help/de/html/ch06s05.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Globale Variablen und
Variablenbereiche</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Genius-Handbuch"><link rel="up" href="ch06.html" title="Kapitel 6. Programmierung
mit GEL"><link rel="prev" href="ch06s04.html" title="Vergleichsoperatoren"><link rel="next"
href="ch06s06.html" title="Parametervariablen"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Globale Variablen und Variablenbereiche</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Zurück</a> </td><th width="60%"
align="center">Kapitel 6. Programmierung mit GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><
div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Globale Variablen und Variablenbereiche</h2></div></div></div><p>
GEL is a
- <a class="ulink" href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
dynamically scoped language</a>. We will explain what this
means below. That is, normal variables and functions are dynamically
scoped. The exception are
diff --git a/help/de/html/ch07s02.html b/help/de/html/ch07s02.html
index 81d8632..7488804 100644
--- a/help/de/html/ch07s02.html
+++ b/help/de/html/ch07s02.html
@@ -3,10 +3,32 @@
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
- as just sequence of lines as if were entered on the command line.
+ as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
- parentheses).
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
</p><p>
The following code will produce an error when entered on the top
level of a program, while it will work just fine in a function.
diff --git a/help/de/html/ch11s04.html b/help/de/html/ch11s04.html
index 3f11173..cc2a2c9 100644
--- a/help/de/html/ch11s04.html
+++ b/help/de/html/ch11s04.html
@@ -2,26 +2,26 @@
Catalan's Constant, approximately 0.915... It is defined to be the series where terms are
<strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, where <code class="varname">k</code>
ranges from 0 to infinity.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Aliase: <code class="function">gamma</code></p><p>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>Der
»Goldene Schnitt«.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
round and uniform.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
is the exponential function
@@ -30,7 +30,7 @@
several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>
@@ -38,7 +38,7 @@
to its diameter. This is approximately 3.14159265359...
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Zurück</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Nach oben</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s05.html">Weiter</a></td></tr><tr><td width="40%" align="left" valign="top">Parameter
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Zum Anfang</a></td><td width="40%"
align="right" valign="top"> Numerik</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s05.html b/help/de/html/ch11s05.html
index 2351192..63c6cf3 100644
--- a/help/de/html/ch11s05.html
+++ b/help/de/html/ch11s05.html
@@ -5,7 +5,7 @@
to <strong class="userinput"><code>|x|</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
<a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
<a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
@@ -14,16 +14,16 @@ for more information.
</p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Ersetzung sehr kleiner Zahlen durch Null.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Aliases: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calculates the complex conjugate of the complex number <code
class="varname">z</code>. If <code class="varname">z</code> is a vector or matrix,
all its elements are conjugated.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Get the denominator of a rational number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Return the fractional part of a number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Aliases: <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Division ohne Rest.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
<strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
<strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Check if argument is a possibly complex rational number.
That is, if both real and imaginary parts are
@@ -32,10 +32,10 @@ all its elements are conjugated.</p><p>
are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Check if argument is an integer (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Aliases: <code
class="function">IsNaturalNumber</code></p><p>Check if argument is a positive real integer. Note that
we accept the convention that 0 is not a natural number.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Check if argument is a rational number (non-complex). Of course rational simply means "not
stored as a floating point number."</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (num)</pre><p>Check if
argument is a real number.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Get
the numerator of a rational number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Aliases: <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Aliases: <code class="function">sign</code></p><p>Return the sign of a
number. That is returns
<code class="literal">-1</code> if value is negative,
<code class="literal">0</code> if value is zero and
@@ -61,12 +61,12 @@ value then <code class="function">Sign</code> returns the direction or 0.
logarithm</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Make number a floating point value. That is returns the floating point
representation of the number <code class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Aliases: <code
class="function">Floor</code></p><p>Get the highest integer less than or equal to <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>The natural logarithm, the logarithm to base <code
class="varname">e</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logarithm of <code
class="varname">x</code> base <code class="varname">b</code> (calls <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> if in modulo
mode), if base is not given, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> is used.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logarithm of <code
class="varname">x</code> base 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Aliases: <code
class="function">lg</code></p><p>Logarithm of <code class="varname">x</code> base 2.</p></dd><dt><span
class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synop
sis">max (a,args...)</pre><p>Aliases: <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Returns the maximum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,args...)</pre><p>Aliases: <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Returns the minimum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(size...)</pre><p>Generate random float in the range <code class="literal">[0,1)</code>.
diff --git a/help/de/html/ch11s06.html b/help/de/html/ch11s06.html
index 2422b86..6d1b317 100644
--- a/help/de/html/ch11s06.html
+++ b/help/de/html/ch11s06.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometrie</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius-Handbuch"><link rel="up" href="ch11.html"
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header"><tr><th colspan="3" align="center">Trigonometrie</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11. Liste der
GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-trigonometry"></a>Trigonometrie</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Aliases: <code
class="function">arccos</code></p><p>The arccos (inverse cos) function.</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Aliases: <code
class="function">arccosh</code></p><p>The arccosh (inverse cosh) function.</p></dd><dt><span class="term"><a
name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Aliases: <code
class="function">arccot</code></p><p>The arccot (inverse cot) function.</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Aliases: <code
class="function">arccoth</code></p><p>The arccoth (inverse coth) function.</p><
/dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc
(x)</pre><p>Aliases: <code class="function">arccsc</code></p><p>The inverse cosecant
function.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Aliases: <code class="function">arccsch</code></p><p>The inverse
hyperbolic cosecant function.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Aliases: <code
class="function">arcsec</code></p><p>The inverse secant function.</p></dd><dt><span class="term"><a
name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech (x)</pre><p>Aliases: <code
class="function">arcsech</code></p><p>The inverse hyperbolic secant function.</p></dd><dt><span
class="term"><a name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin
(x)</pre><p>Aliases: <code class="fun
ction">arcsin</code></p><p>The arcsin (inverse sin) function.</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Aliases: <code
class="function">arcsinh</code></p><p>The arcsinh (inverse sinh) function.</p></dd><dt><span class="term"><a
name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Aliases: <code
class="function">arctan</code></p><p>Calculates the arctan (inverse tan) function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Aliases: <code class="function">arctanh</code></p><p>The arctanh (inverse
tanh) function.</p></dd><dt><span class="term"><a name="gel-function-atan2"></a>atan2</span></dt><dd><pre
class="synopsis">atan2 (y, x)</pre><p>Aliases: <code class="function">arctan2</code></p><p>Calculates the
arctan2 function. If
<strong class="userinput"><code>x>0</code></strong> then it returns
@@ -11,11 +11,11 @@
rather than failing.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Calculates the cosine function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Calculates the hyperbolic cosine function.</p><p>
See
@@ -23,7 +23,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>The cotangent function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>The hyperbolic cotangent function.</p><p>
See
@@ -31,7 +31,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>The cosecant function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>The hyperbolic cosecant function.</p><p>
See
@@ -39,7 +39,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>The secant function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>The hyperbolic secant function.</p><p>
See
@@ -47,7 +47,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Calculates the sine function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Calculates the hyperbolic sine function.</p><p>
See
@@ -55,7 +55,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calculates the tan function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>The hyperbolic tangent function.</p><p>
See
diff --git a/help/de/html/ch11s07.html b/help/de/html/ch11s07.html
index 7c0b57a..28b700d 100644
--- a/help/de/html/ch11s07.html
+++ b/help/de/html/ch11s07.html
@@ -8,14 +8,14 @@
<a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Return the <code class="varname">n</code>th Bernoulli number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Aliases: <code class="function">CRT</code></p><p>Find the
<code class="varname">x</code> that solves the system given by
the vector <code class="varname">a</code> and modulo the elements of
<code class="varname">m</code>, using the Chinese Remainder Theorem.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Given two factorizations, give the factorization of the
@@ -23,7 +23,7 @@
F<sub>q</sub>, the finite field of order <code class="varname">q</code>, where <code
class="varname">q</code>
is a prime, using the Silver-Pohlig-Hellman algorithm.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Checks divisibility (if <code class="varname">m</code> divides <code
class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>
@@ -32,7 +32,7 @@
relatively prime to <code class="varname">n</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>
@@ -52,7 +52,7 @@
1 2 1]</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>
Return all factors of <code class="varname">n</code> in a vector. This
includes all the non-prime factors as well. It includes 1 and the
@@ -75,7 +75,7 @@
of two factors that are very close to each other.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Find the first primitive element in F<sub>q</sub>, the
finite
group of order <code class="varname">q</code>. Of course <code class="varname">q</code> must be a
prime.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Find a random primitive element in F<sub>q</sub>,
the finite
group of order <code class="varname">q</code> (q must be a prime).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of n in F<sub>q</sub>, the finite
@@ -99,7 +99,7 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -112,8 +112,8 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.
</p></dd><dt><span class="term"><a name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre
class="synopsis">IsOdd (n)</pre><p>Überprüft, ob eine Ganzzahl ungerade ist.</p></dd><dt><span
class="term"><a name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre
class="synopsis">IsPerfectPower (n)</pre><p>Check an integer for being any perfect power,
a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
Check an integer for being a perfect square of an integer. The number must
- be a real integer. Negative integers are of course never perfect
- squares of real integers.
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
</p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>
Tests primality of integers, for numbers less than 2.5e10 the
answer is deterministic (if Riemann hypothesis is true). For
@@ -151,12 +151,12 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returns the <code class="varname">n</code>th Lucas number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Return all maximal prime power factors of a
number.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>
@@ -170,7 +170,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -185,7 +185,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
better on smaller integers.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
@@ -194,7 +194,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
result is deterministic.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Returns inverse of n mod m.</p><p>
diff --git a/help/de/html/ch11s08.html b/help/de/html/ch11s08.html
index 5a18e93..af17c56 100644
--- a/help/de/html/ch11s08.html
+++ b/help/de/html/ch11s08.html
@@ -1,11 +1,11 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Matrixoperationen</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius-Handbuch"><link rel="up" href="ch11.html"
title="Kapitel 11. Liste der GEL-Funktionen"><link rel="prev" href="ch11s07.html" title="Zahlentheorie"><link
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header"><tr><th colspan="3" align="center">Matrixoperationen</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s07.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11. Liste der
GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s09.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title
" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Matrixoperationen</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Apply a function over all entries of a matrix and return a matrix of the
results.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Apply a function over all entries of 2 matrices (or 1
value and 1 matrix) and return a matrix of the results.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Gets
the columns of a matrix as a horizontal vector.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre class="synops
is">ComplementSubmatrix (m,r,c)</pre><p>Remove column(s) and row(s) from a matrix.</p></dd><dt><span
class="term"><a name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre
class="synopsis">CompoundMatrix (k,A)</pre><p>Calculate the kth compound matrix of A.</p></dd><dt><span
class="term"><a name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
- Count the number of zero columns in a matrix. For example
- once your column reduce a matrix you can use this to find
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
</p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Löscht eine Spalte einer Matrix.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Löscht eine Zeile einer Matrix.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Gets the diagonal entries of a matrix as a column vector.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
See
@@ -28,7 +28,7 @@
<strong class="userinput"><code>5</code></strong>, we return <strong
class="userinput"><code>[1,4,5]</code></strong>. If
<code class="varname">msize</code> is 0, we always return <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Is
a matrix diagonal.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Check if a matrix is the identity matrix. Automatically returns <code
class="constant">false</code>
if the matrix is not square. Also works on numbers, in which
@@ -37,12 +37,12 @@
no error is generated and <code class="constant">false</code> is returned.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Is a matrix lower triangular. That is, are all the entries
above the diagonal zero.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Check if a matrix is non-negative, that is if each element
is non-negative.
Do not confuse positive matrices with positive semi-definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Check if a matrix is positive, that is if each element is
positive (and hence real). In particular, no element is 0. Do not confuse
positive matrices with positive definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Check if a matrix is a matrix of rational (non-complex)
numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Check if a matrix is a matrix of real (non-complex) numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>
Check if a matrix is square, that is its width is equal to
@@ -62,7 +62,7 @@ functions make this check. Values can be any number including complex numbers.<
<strong class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> is the same as
<strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Make column vector out of matrix by putting columns above
each other. Returns <code class="constant">null</code> when given <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>
diff --git a/help/de/html/ch11s09.html b/help/de/html/ch11s09.html
index f046e0d..37bae21 100644
--- a/help/de/html/ch11s09.html
+++ b/help/de/html/ch11s09.html
@@ -50,7 +50,7 @@ result as a vector and not added together.</p></dd><dt><span class="term"><a nam
diagonal).
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Get the eigenvectors of a square matrix. Optionally get also
@@ -58,7 +58,7 @@ the eigenvalues and their algebraic multiplicities.
Currently only works for matrices of size up to 2 by 2.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Apply the Gram-Schmidt process (to the columns) with respect to
@@ -152,7 +152,7 @@ determinant.
of two matrices.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
@@ -182,7 +182,7 @@ determinant.
and <code class="varname">U</code> to <code class="constant">null</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Get the <code class="varname">i</code>-<code class="varname">j</code>
minor of a matrix.</p><p>
@@ -218,7 +218,7 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<code class="varname">Q</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Return the Rayleigh quotient (also called the Rayleigh-Ritz
quotient or ratio) of a matrix and a vector.</p><p>
@@ -241,45 +241,45 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</p></dd><dt><span
class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Return the matrix corresponding
to rotation around origin in R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
x-axis.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (angle)</pre><p>Return the matrix corresponding to rotation around origin in
R<sup>3</sup> about the
y-axis.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
z-axis.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Get a basis matrix for the rowspace of a matrix.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Evaluate (v,w) with respect to the sesquilinear form given
by the matrix A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Return a function that evaluates two vectors with
respect to the sesquilinear form given by A.</p></dd><dt><span class="term"><a name="gel
-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returns the Smith normal form of a matrix over fields (will
end up with 1's on the diagonal).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Solve linear system Mx=V, return solution V if there
is a unique solution, <code class="constant">null</code> otherwise. Extra two reference parameters can
optionally be used to get the reduced M and V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Return the Toeplitz matrix constructed given the first column c
and (optionally) the first row r. If only the column c is given then it is
conjugated and the nonconjugated version is used for the first row to give a
Hermitian matrix (if the first element is real of course).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Aliases: <code class="function">trace</code></p><p>Calculate the trace of
a matrix. That is the sum of the diagonal elements.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Transpose of a matrix. This is the same as the
<strong class="userinput"><code>.'</code></strong> operator.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Aliases: <code class="function">vander</code></p><p>Return the
Vandermonde matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>The angle of two vectors with respect to inner product given by
<code class="varname">B</code>. If <code class="varname">B</code> is not given then the standard
Hermitian product is used. <code class="varname">B</code> can either be a sesquilinear
function of two arguments or it can be a matrix giving a sesquilinear form.
</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>The direct sum of the vector spaces M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersection of the subspaces given by M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>The sum of the vector spaces M and N, that is {w | w=m+n, m
in M, n in N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Aliases: <code class="function">Adjugate</code></p><p>Get the classical
adjoint (adjugate) of a matrix.</p></dd><dt><span class="term"><a name="gel-function-cref"></a>cref</spa
n></dt><dd><pre class="synopsis">cref (M)</pre><p>Aliases: <code class="function">CREF</code> <code
class="function">ColumnReducedEchelonForm</code></p><p>Compute the Column Reduced Echelon
Form.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Aliases: <code class="function">Determinant</code></p><p>Get the determinant
of a matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Aliases: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Get the row echelon form of a matrix. That is, apply gaussian
elimination but not backaddition to <code class="varname">M</code>. The pivot rows are
divided to make all pivots 1.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Aliases: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Get the reduced row echelon form of a matrix. That is,
apply gaussian elimination together with backaddition to <code class="varname">M</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s08.html">Zurück</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Nach oben</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s10.html">Weiter</a></td></tr><tr><td width="40%" align="left"
valign="top">Matrixoperationen </td><td width="20%" align="center"><a accesskey="h" href="index.html">Zum
Anfang</a></td><td width="40%" align="right" valign="top"> Kombinatorik</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s10.html b/help/de/html/ch11s10.html
index bff4ffe..0ae020f 100644
--- a/help/de/html/ch11s10.html
+++ b/help/de/html/ch11s10.html
@@ -3,7 +3,10 @@
<a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Get all combinations of k numbers from 1 to n as a vector of vectors.
(See also <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)
-</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
See
<a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Factorial: <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
@@ -20,17 +23,18 @@
<strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
- Calculate the Frobenius number. That is calculate smallest
+ Calculate the Frobenius number. That is calculate largest
number that cannot be given as a non-negative integer linear
combination of a given vector of non-negative integers.
The vector can be given as separate numbers or a single vector.
All the numbers given should have GCD of 1.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(combining_rule)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>
Find the vector <code class="varname">c</code> of non-negative integers
@@ -40,8 +44,18 @@
of non-negative integers.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coeffi
cients. Takes a vector of
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coefficients. Takes a vector of
<code class="varname">k</code>
non-negative integers and computes the multinomial coefficient.
This corresponds to the coefficient in the homogeneous polynomial
@@ -57,7 +71,7 @@
<strong class="userinput"><code>Binomial(a+b,b)</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Get combination that would come after v in call to
@@ -77,6 +91,9 @@ do (
) while not IsNull(n:=NextCombination(n,6));</code></strong>
</pre><p>
See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Get the Pascal's triangle as a matrix. This will return
an <code class="varname">i</code>+1 by <code class="varname">i</code>+1 lower diagonal
matrix that is the Pascal's triangle after <code class="varname">i</code>
@@ -86,7 +103,7 @@ do (
</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Get all permutations of <code class="varname">k</code> numbers from 1 to <code
class="varname">n</code> as a vector of vectors.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Aliases: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k =
n(n+1)...(n+(k-1)).</p><p>
See
<a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
@@ -109,5 +126,5 @@ do (
<code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Zurück</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Nach oben</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">Weiter</a></td></tr><tr><td width="40%" align="left" valign="top">Lineare
Algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Zum Anfang</a></td><td
width="40%" align="right" valign="top"> Analysis</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s11.html b/help/de/html/ch11s11.html
index 6fad0ca..9cbc284 100644
--- a/help/de/html/ch11s11.html
+++ b/help/de/html/ch11s11.html
@@ -25,7 +25,7 @@ the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, whil
<strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Either <code class="varname">a</code>
or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Try to calculate an infinite product for a single parameter
function.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Try to calculate an infinite product for a
double parameter function with func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Try to calculate an infinite sum for a single parameter function.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,inc)</pre><p>Try to calculate an infinite sum for a double
parameter function with func(arg,n).</p></dd><d
t><span class="term"><a name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre
class="synopsis">IsContinuous (f,x0)</pre><p>Try and see if a real-valued function is continuous at x0 by
calculating the limit there.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Test for differentiability by approximating the left and
right limits and comparing.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calculate the left limit of a real-valued function at x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Calculate the
limit of a real-valued function at x0. Tries to calculate both left and right limits.</p></dd><dt><span
class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</sp
an></dt><dd><pre class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integration by midpoint
rule.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Aliases: <code
class="function">NDerivative</code></p><p>Attempt to calculate numerical derivative.</p><p>
See
@@ -40,7 +40,7 @@ up to <code class="varname">N</code>th harmonic computed numerically. The coeff
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
@@ -50,7 +50,7 @@ trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the cosine Fourier series of
@@ -65,7 +65,7 @@ Note that <strong class="userinput"><code>a@(1)</code></strong> is
the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -76,7 +76,7 @@ only has cosine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the sine Fourier series of
@@ -88,7 +88,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -99,7 +99,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration by rule set in NumericalIntegralFunction of f
from a to b using NumericalIntegralSteps steps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Attempt to calculate numerical left
derivative.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Attempt to
calculate the limit of f(step_fun(i)) as i goes from 1 to N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">Nume
ricalRightDerivative (f,x0)</pre><p>Attempt to calculate numerical right derivative.</p></dd><dt><span
class="term"><a name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
<code class="function">f</code> with half period <code class="varname">L</code>. That
diff --git a/help/de/html/ch11s12.html b/help/de/html/ch11s12.html
index f4f05f4..d3b158e 100644
--- a/help/de/html/ch11s12.html
+++ b/help/de/html/ch11s12.html
@@ -1,21 +1,21 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Funktionen</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius-Handbuch"><link rel="up" href="ch11.html"
title="Kapitel 11. Liste der GEL-Funktionen"><link rel="prev" href="ch11s11.html" title="Analysis"><link
rel="next" href="ch11s13.html" title="Gleichungen lösen"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Funktionen</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s11.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11. Liste der
GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s13.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-functions"></a>Funktionen</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Aliases:
<code class="function">Arg</code> <code class="function">arg</code></p><p>argument (angle) of complex
number.</p></dd><dt><span class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre
class="synopsis">BesselJ0 (x)</pre><p>Bessel function of the first kind of order 0. Only implemented for
real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Bessel
function of the first kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Bessel
function of the first kind of order <code class="varname">n</code>. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Bessel
function of the second kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Bessel
function of the second kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Bessel
function of the second kind of order <code class="varname">n</code>. Only implemented for real
numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Aliases: <code class="function">erf</code></p><p>The error function, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
@@ -27,7 +27,7 @@
</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Aliases: <code class="function">Gamma</code></p><p>The Gamma function. Currently only
implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returns 1 if and only if all elements are equal.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>
The principal branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>.
@@ -38,7 +38,7 @@
See <a class="link" href="ch11s12.html#gel-function-LambertWm1"><code
class="function">LambertWm1</code></a> for the other real branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>
The minus-one branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>
@@ -48,29 +48,34 @@
See <a class="link" href="ch11s12.html#gel-function-LambertW"><code
class="function">LambertW</code></a> for the principal branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Find the first value where f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Moebius mapping of the disk to itself mapping a to 0.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity
respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Moebius mapping using the cross ratio taking
infinity to infinity and z2,z3 to 1 and 0 respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 1 and z3,z4 to 0 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 0 and z2,z4 to 1 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poisson kernel on D(0,R) (not normalized to
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Aliases: <code class="function">zeta</code></p><p>The Riemann zeta
function. Currently only implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>The unit step function is 0 for x<0, 1 otherwise. This is the
integral of the Dirac Delta function. Also called the Heaviside function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>
The <code class="function">cis</code> function, that is the same as
<strong class="userinput"><code>cos(x)+1i*sin(x)</code></strong>
@@ -78,5 +83,5 @@
<strong class="userinput"><code>sin(x)/x</code></strong>.
If you want the normalized function call <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
</p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Zurück</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Nach
oben</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s13.html">Weiter</a></td></tr><tr><td
width="40%" align="left" valign="top">Analysis </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Zum Anfang</a></td><td width="40%" align="right" valign="top"> Gleichungen
lösen</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s13.html b/help/de/html/ch11s13.html
index aef0113..c682bf5 100644
--- a/help/de/html/ch11s13.html
+++ b/help/de/html/ch11s13.html
@@ -10,7 +10,7 @@
See
<a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
@@ -29,12 +29,12 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
@@ -73,7 +73,7 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Find root of a function using the bisection method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
@@ -102,7 +102,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Find zeros using Newton's method. <code class="varname">f</code> is
the function and <code class="varname">df</code> is the derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
@@ -116,7 +116,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>
Compute roots of a polynomial (degrees 1 through 4)
using one of the formulas for such polynomials.
@@ -139,8 +139,9 @@
Returns a column vector of the two solutions.
</p><p>
See
- <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>
Compute roots of a quartic (degree 4) polynomial using the
quartic formula. The polynomial should be given as a
@@ -152,7 +153,7 @@
See
<a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
@@ -168,14 +169,14 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
@@ -209,5 +210,5 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Zurück</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Nach
oben</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s14.html">Weiter</a></td></tr><tr><td
width="40%" align="left" valign="top">Funktionen </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Zum Anfang</a></td><td width="40%" align="right" valign="top">
Statistik</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s14.html b/help/de/html/ch11s14.html
index fd3dea1..e0beddf 100644
--- a/help/de/html/ch11s14.html
+++ b/help/de/html/ch11s14.html
@@ -1,20 +1,27 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Statistik</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Genius-Handbuch"><link rel="up" href="ch11.html" title="Kapitel 11. Liste der GEL-Funktionen"><link
rel="prev" href="ch11s13.html" title="Gleichungen lösen"><link rel="next" href="ch11s15.html"
title="Polynomials"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Statistik</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s13.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11. Liste der
GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-statistics"></a>Statistik</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average of an entire matrix.</p><p>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Statistik</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Genius-Handbuch"><link rel="up" href="ch11.html" title="Kapitel 11. Liste der GEL-Funktionen"><link
rel="prev" href="ch11s13.html" title="Gleichungen lösen"><link rel="next" href="ch11s15.html"
title="Polynomials"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Statistik</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s13.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11. Liste der
GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-statistics"></a>Statistik</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral of the GaussFunction from 0 to <code
class="varname">x</code> (area under the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>The normalized Gauss distribution function (the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Aliases: <code class="function">median</code></p><p>Calculate median of
an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calculate median of each row in a matrix and return a column
vector of the medians.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdevp</code></p><p>Calculate the population standard deviations of rows of a matrix and
return a vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdev</code></p><p>Calculate the standard deviations of rows of a matrix and return a
vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Aliases: <code class="function">stdev</code></p><p>Calculate
the standard deviation of a whole matrix.</p></dd></dl></div></div><div class="navfooter"><hr><table widt
h="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Zurück</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Nach
oben</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s15.html">Weiter</a></td></tr><tr><td
width="40%" align="left" valign="top">Gleichungen lösen </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Zum Anfang</a></td><td width="40%" align="right" valign="top">
Polynomials</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s15.html b/help/de/html/ch11s15.html
index 73e917c..67cb7cc 100644
--- a/help/de/html/ch11s15.html
+++ b/help/de/html/ch11s15.html
@@ -17,5 +17,5 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Take second polynomial (as vector)
derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Take polynomial (as vector) derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Make function out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Make string out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Subtract two polynomials (as vectors).
</p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Trim zeros from a polynomial (as vector).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Zurück</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Nach oben</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Weiter</a></td></tr><tr><td width="40%" align="left" valign="top">Statistik </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Zum Anfang</a></td><td width="40%"
align="right" valign="top"> Mengenlehre</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s18.html b/help/de/html/ch11s18.html
index 6c12478..742bd40 100644
--- a/help/de/html/ch11s18.html
+++ b/help/de/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Verschiedenes</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius-Handbuch"><link rel="up" href="ch11.html"
title="Kapitel 11. Liste der GEL-Funktionen"><link rel="prev" href="ch11s17.html" title="Commutative
Algebra"><link rel="next" href="ch11s19.html" title="Symbolische Operationen"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Verschiedenes</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11.
Liste der GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class=
"title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Verschiedenes</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a vector of ASCII values.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Conve
rt a string to a vector of 0-based alphabet values (positions in the alphabet string), -1's for unknown
letters.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Zurück</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Nach oben</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Weiter</a></td></tr><tr><td width="40%" align="left"
valign="top">Commutative Algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Zum
Anfang</a></td><td width="40%" align="right" valign="top"> Symbolische
Operationen</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Verschiedenes</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Genius-Handbuch"><link rel="up" href="ch11.html"
title="Kapitel 11. Liste der GEL-Funktionen"><link rel="prev" href="ch11s17.html" title="Commutative
Algebra"><link rel="next" href="ch11s19.html" title="Symbolische Operationen"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Verschiedenes</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Zurück</a> </td><th width="60%" align="center">Kapitel 11.
Liste der GEL-Funktionen</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Weiter</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class=
"title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Verschiedenes</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Zurück</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Nach oben</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Weiter</a></td></tr><tr><td width="40%" align="left"
valign="top">Commutative Algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Zum
Anfang</a></td><td width="40%" align="right" valign="top"> Symbolische
Operationen</td></tr></table></div></body></html>
diff --git a/help/de/html/ch11s20.html b/help/de/html/ch11s20.html
index 4bdc146..7916d42 100644
--- a/help/de/html/ch11s20.html
+++ b/help/de/html/ch11s20.html
@@ -102,7 +102,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
</pre><p>
@@ -153,7 +153,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
</pre><p>
@@ -330,7 +330,7 @@ limits as <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.
<code class="varname">n</code> by 3 matrix for a longer polyline.
</p><p>
Extra parameters can be added to specify line color, thickness,
- arrows, the plotting window, or legend.
+ the plotting window, or legend.
You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
<strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>,
diff --git a/help/de/html/index.html b/help/de/html/index.html
index 64f53cb..f76acf6 100644
--- a/help/de/html/index.html
+++ b/help/de/html/index.html
@@ -1,4 +1,4 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Genius-Handbuch</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><meta name="description" content="Handbuch für das Genius Mathematikwerkzeug."><link rel="home"
href="index.html" title="Genius-Handbuch"><link rel="next" href="ch01.html" title="Kapitel 1.
Einführung"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Genius-Handbuch</th></tr><tr><td width="20%" align="left"> </td><th width="60%"
align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Weiter</a></td></tr></table><hr></div><div lang="de" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Genius-Handbuch</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><span class="fi
rstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Oklahoma State University<br></span><div class="address"><p> <code class="email"><<a
class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code> </p></div></div></div><div
class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">University of Queensland,
Australien<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:kaiw itee
uq edu au">kaiw itee uq edu au</a>></code> </p></div></div></div></div></div><div><p
class="releaseinfo">This manual describes version 1.0.22 of Genius.
- </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2009, 2011
Mario Blättermann (mariobl freenet de)</p></div><div><div class="legalnotice"><a
name="legalnotice"></a><p>Das vorliegende Dokument kann gemäß den Bedingungen der GNU Free Documentation
License (GFDL), Version 1.1 oder jeder späteren, von der Free Software Foundation veröffentlichten Version
ohne unveränderbare Abschnitte sowie ohne Texte auf dem vorderen und hinteren Buchdeckel kopiert, verteilt
und/oder modifiziert werden. Eine Kopie der GFDL finden Sie unter diesem <a class="ulink" href="ghelp:fdl"
target="_top">Link</a> oder in der mit diesem Handbuch gelieferten Datei COPYING-DOCS.</p><p>Dieses Handbuch
ist Teil einer Sammlung von GNOME-Handbüchern, die unter der GFDL veröffentlicht werden. Wenn Sie dieses
Handbuch getrennt von der Sammlung weiterverbreite
n möchten, können Sie das tun, indem Sie eine Kopie der Lizenz zum Handbuch hinzufügen, wie es in Abschnitt
6 der Lizenz beschrieben ist.</p><p>Viele der Namen, die von Unternehmen verwendet werden, um ihre Produkte
und Dienstleistungen von anderen zu unterscheiden, sind eingetragene Warenzeichen. An den Stellen, an denen
diese Namen in einer GNOME-Dokumentation erscheinen, werden die Namen in Großbuchstaben oder mit einem großen
Anfangsbuchstaben geschrieben, wenn das GNOME-Dokumentationsprojekt auf diese Warenzeichen hingewiesen
wird.</p><p>DAS DOKUMENT UND VERÄNDERTE FASSUNGEN DES DOKUMENTS WERDEN UNTER DEN BEDINGUNGEN DER GNU FREE
DOCUMENTATION LICENSE ZUR VERFÜGUNG GESTELLT MIT DEM WEITERGEHENDEN VERSTÄNDNIS, DASS: </p><div
class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>DIESES DOKUMENT WIRD »WIE
VORLIEGEND« GELIEFERT, OHNE GARANTIEN IRGENDEINER ART, SOWOHL AUSDRÜCKLICH GENANNTE ALS AUCH ANGEDEUTETE.
DIES BEZIEHT SICH AUCH OHNE EI
NSCHRÄNKUNG AUF GARANTIEN, DASS DIESES DOKUMENT ODER VERÄNDERTE FASSUNGEN DIESES DOKUMENTS FREI VON
HANDELSDEFEKTEN, FÜR EINEN BESTIMMTEN ZWECK GEEIGNET IST ODER DASS ES KEINE RECHTE DRITTER VERLETZT. DAS
VOLLE RISIKO WAS QUALITÄT, GENAUIGKEIT UND LEISTUNG DES DOKUMENTS ODER VERÄNDERTE FASSUNGEN DES DOKUMENTS
LIEGT BEI IHNEN. SOLLTE EIN DOKUMENT ODER EINE VERÄNDERTE FASSUNG DAVON FEHLER IRGENDEINER ART BEINHALTEN,
TRAGEN SIE (NICHT DER URSPRUNGSAUTOR, DER AUTOR ODER EIN MITWIRKENDER) DIE KOSTEN FÜR NOTWENDIGE
DIENSTLEISTUNGEN, REPARATUREN ODER FEHLERKORREKTUREN. DIESER HAFTUNGSAUSSCHLUSS IST EIN ESSENZIELLER TEIL
DIESER LIZENZ. DIE VERWENDUNG EINES DOKUMENTS ODER EINER VERÄNDERTEN VERSION DES DOKUMENTS IST NICHT
GESTATTET AUßER UNTER BEACHTUNG DIESES HAFTUNGSAUSSCHLUSSES UND</p></li><li class="listitem"><p>UNTER KEINEN
UMSTÄNDEN UND AUF BASIS KEINER RECHTSGRUNDLAGE, EGAL OB DURCH UNERLAUBTEN HANDLUNGEN (EINSCHLIEßLICH
FAHRLÄSSIGKEIT), VERTRAG ODER ANDERWEITIG KAN
N DER AUTOR, URSPRUNGSAUTOR, EIN MITWIRKENDER ODER EIN VERTRIEBSPARTNER DIESES DOKUMENTS ODER EINER
VERÄNDERTEN FASSUNG DES DOKUMENTS ODER EIN ZULIEFERER EINER DIESER PARTEIEN, HAFTBAR GEMACHT WERDEN FÜR
DIREKTE, INDIREKTE, SPEZIELLE, VERSEHENTLICHE ODER FOLGESCHÄDEN JEGLICHER ART, EINSCHLIEßLICH UND OHNE
EINSCHRÄNKUNGEN SCHÄDEN DURCH VERLUST VON KULANZ, ARBEITSAUSFALL, COMPUTERVERSAGEN ODER
COMPUTERFEHLFUNKTIONEN ODER ALLE ANDEREN SCHÄDEN ODER VERLUSTE, DIE SICH AUS ODER IN VERBINDUNG MIT DER
VERWENDUNG DES DOKUMENTS UND VERÄNDERTER FASSUNGEN DES DOKUMENTS ERGEBEN, AUCH WENN DIE OBEN GENANNTEN
PARTEIEN ÜBER DIE MÖGLICHKEIT SOLCHER SCHÄDEN INFORMIERT WAREN.</p></li></ol></div></div></div><div><div
class="legalnotice"><a name="idm46132110458992"></a><p
class="legalnotice-title"><b>Rückmeldungen</b></p><p>Um einen Fehler zu melden oder einen Vorschlag zur
Anwendung <span class="application">Genius Mathematikwerkzeug</span> oder zu diesem Handbuch zu machen, folge
S
ie den Anweisungen auf der <a class="ulink" href="http://www.jirka.org/genius.html";
target="_top">Genius-Webseite</a> oder schreiben Sie eine E-Mail an <code class="email"><<a class="email"
href="mailto:jirka 5z com">jirka 5z com</a>></code>.</p></div></div><div><div class="revhistory"><table
style="border-style:solid; width:100%;" summary="Versionsgeschichte"><tr><th align="left" valign="top"
colspan="2"><b>Versionsgeschichte</b></th></tr><tr><td align="left">Version 0.2</td><td
align="left">September 2016</td></tr><tr><td align="left" colspan="2">
+ </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2009, 2011
Mario Blättermann (mariobl freenet de)</p></div><div><div class="legalnotice"><a
name="legalnotice"></a><p>Das vorliegende Dokument kann gemäß den Bedingungen der GNU Free Documentation
License (GFDL), Version 1.1 oder jeder späteren, von der Free Software Foundation veröffentlichten Version
ohne unveränderbare Abschnitte sowie ohne Texte auf dem vorderen und hinteren Buchdeckel kopiert, verteilt
und/oder modifiziert werden. Eine Kopie der GFDL finden Sie unter diesem <a class="ulink" href="ghelp:fdl"
target="_top">Link</a> oder in der mit diesem Handbuch gelieferten Datei COPYING-DOCS.</p><p>Dieses Handbuch
ist Teil einer Sammlung von GNOME-Handbüchern, die unter der GFDL veröffentlicht werden. Wenn Sie dieses
Handbuch getrennt von der Sammlung weiterverbreite
n möchten, können Sie das tun, indem Sie eine Kopie der Lizenz zum Handbuch hinzufügen, wie es in Abschnitt
6 der Lizenz beschrieben ist.</p><p>Viele der Namen, die von Unternehmen verwendet werden, um ihre Produkte
und Dienstleistungen von anderen zu unterscheiden, sind eingetragene Warenzeichen. An den Stellen, an denen
diese Namen in einer GNOME-Dokumentation erscheinen, werden die Namen in Großbuchstaben oder mit einem großen
Anfangsbuchstaben geschrieben, wenn das GNOME-Dokumentationsprojekt auf diese Warenzeichen hingewiesen
wird.</p><p>DAS DOKUMENT UND VERÄNDERTE FASSUNGEN DES DOKUMENTS WERDEN UNTER DEN BEDINGUNGEN DER GNU FREE
DOCUMENTATION LICENSE ZUR VERFÜGUNG GESTELLT MIT DEM WEITERGEHENDEN VERSTÄNDNIS, DASS: </p><div
class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>DIESES DOKUMENT WIRD »WIE
VORLIEGEND« GELIEFERT, OHNE GARANTIEN IRGENDEINER ART, SOWOHL AUSDRÜCKLICH GENANNTE ALS AUCH ANGEDEUTETE.
DIES BEZIEHT SICH AUCH OHNE EI
NSCHRÄNKUNG AUF GARANTIEN, DASS DIESES DOKUMENT ODER VERÄNDERTE FASSUNGEN DIESES DOKUMENTS FREI VON
HANDELSDEFEKTEN, FÜR EINEN BESTIMMTEN ZWECK GEEIGNET IST ODER DASS ES KEINE RECHTE DRITTER VERLETZT. DAS
VOLLE RISIKO WAS QUALITÄT, GENAUIGKEIT UND LEISTUNG DES DOKUMENTS ODER VERÄNDERTE FASSUNGEN DES DOKUMENTS
LIEGT BEI IHNEN. SOLLTE EIN DOKUMENT ODER EINE VERÄNDERTE FASSUNG DAVON FEHLER IRGENDEINER ART BEINHALTEN,
TRAGEN SIE (NICHT DER URSPRUNGSAUTOR, DER AUTOR ODER EIN MITWIRKENDER) DIE KOSTEN FÜR NOTWENDIGE
DIENSTLEISTUNGEN, REPARATUREN ODER FEHLERKORREKTUREN. DIESER HAFTUNGSAUSSCHLUSS IST EIN ESSENZIELLER TEIL
DIESER LIZENZ. DIE VERWENDUNG EINES DOKUMENTS ODER EINER VERÄNDERTEN VERSION DES DOKUMENTS IST NICHT
GESTATTET AUßER UNTER BEACHTUNG DIESES HAFTUNGSAUSSCHLUSSES UND</p></li><li class="listitem"><p>UNTER KEINEN
UMSTÄNDEN UND AUF BASIS KEINER RECHTSGRUNDLAGE, EGAL OB DURCH UNERLAUBTEN HANDLUNGEN (EINSCHLIEßLICH
FAHRLÄSSIGKEIT), VERTRAG ODER ANDERWEITIG KAN
N DER AUTOR, URSPRUNGSAUTOR, EIN MITWIRKENDER ODER EIN VERTRIEBSPARTNER DIESES DOKUMENTS ODER EINER
VERÄNDERTEN FASSUNG DES DOKUMENTS ODER EIN ZULIEFERER EINER DIESER PARTEIEN, HAFTBAR GEMACHT WERDEN FÜR
DIREKTE, INDIREKTE, SPEZIELLE, VERSEHENTLICHE ODER FOLGESCHÄDEN JEGLICHER ART, EINSCHLIEßLICH UND OHNE
EINSCHRÄNKUNGEN SCHÄDEN DURCH VERLUST VON KULANZ, ARBEITSAUSFALL, COMPUTERVERSAGEN ODER
COMPUTERFEHLFUNKTIONEN ODER ALLE ANDEREN SCHÄDEN ODER VERLUSTE, DIE SICH AUS ODER IN VERBINDUNG MIT DER
VERWENDUNG DES DOKUMENTS UND VERÄNDERTER FASSUNGEN DES DOKUMENTS ERGEBEN, AUCH WENN DIE OBEN GENANNTEN
PARTEIEN ÜBER DIE MÖGLICHKEIT SOLCHER SCHÄDEN INFORMIERT WAREN.</p></li></ol></div></div></div><div><div
class="legalnotice"><a name="idm51"></a><p class="legalnotice-title"><b>Rückmeldungen</b></p><p>Um einen
Fehler zu melden oder einen Vorschlag zur Anwendung <span class="application">Genius
Mathematikwerkzeug</span> oder zu diesem Handbuch zu machen, folge Sie den Anwei
sungen auf der <a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius-Webseite</a>
oder schreiben Sie eine E-Mail an <code class="email"><<a class="email" href="mailto:jirka 5z com">jirka
5z com</a>></code>.</p></div></div><div><div class="revhistory"><table style="border-style:solid;
width:100%;" summary="Versionsgeschichte"><tr><th align="left" valign="top"
colspan="2"><b>Versionsgeschichte</b></th></tr><tr><td align="left">Version 0.2</td><td
align="left">September 2016</td></tr><tr><td align="left" colspan="2">
<p class="author">Jiri (George) Lebl <code class="email"><<a class="email"
href="mailto:jirka 5z com">jirka 5z com</a>></code></p>
</td></tr></table></div></div><div><div class="abstract"><p
class="title"><b>Zusammenfassung</b></p><p>Handbuch für das Genius
Mathematikwerkzeug.</p></div></div></div><hr></div><div class="toc"><p><b>Inhaltsverzeichnis</b></p><dl
class="toc"><dt><span class="chapter"><a href="ch01.html">1. Einführung</a></span></dt><dt><span
class="chapter"><a href="ch02.html">2. Erste Schritte</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch02.html#genius-to-start"><span class="application">Genius Mathematikwerkzeug
starten</span></a></span></dt><dt><span class="sect1"><a href="ch02s02.html">Beim Start von <span
class="application">Genius</span></a></span></dt></dl></dd><dt><span class="chapter"><a href="ch03.html">3.
Grundlagen der Benutzung</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch03.html#genius-usage-workarea">Benutzung des Arbeitsplatzes</a></span></dt><dt><span class="sect1"><a
href="ch03s02.html">Erstellen eines neuen Programms</a></span></dt><dt><span
class="sect1"><a href="ch03s03.html">Öffnen und Ausführen eines Programms</a></span></dt></dl></dd><dt><span
class="chapter"><a href="ch04.html">4. Darstellung</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch04.html#genius-line-plots">Kurvendarstellungen</a></span></dt><dt><span class="sect1"><a
href="ch04s02.html">Parametrische Darstellungen</a></span></dt><dt><span class="sect1"><a
href="ch04s03.html">Richtungsfeld-Darstellungen</a></span></dt><dt><span class="sect1"><a
href="ch04s04.html">Vektorfeld-Darstellungen</a></span></dt><dt><span class="sect1"><a
href="ch04s05.html">2D-Darstellungen</a></span></dt></dl></dd><dt><span class="chapter"><a
href="ch05.html">5. GEL-Grundlagen</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch05.html#genius-gel-values">Werte</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-numbers">Zahlen</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-booleans">Wahrheitsw
erte</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-strings">Strings</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-null">Null</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s02.html">Verwendung von Variablen</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-setting">Setzen von Variablen</a></span></dt><dt><span
class="sect2"><a href="ch05s02.html#genius-gel-variables-built-in">Eingebaute
Variablen</a></span></dt><dt><span class="sect2"><a href="ch05s02.html#genius-gel-previous-result">Vorherige
Ergebnisvariable</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch05s03.html">Verwendung von
Funktionen</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-defining">Definieren von Funktionen</a></span></dt><dt><span
class="sect2"><a href="ch05s03.html#genius-gel-functions-variable-argument-lists">Variable Argument Lists</a>
</span></dt><dt><span class="sect2"><a href="ch05s03.html#genius-gel-functions-passing-functions">Übergabe
von Funktionen an Funktionen</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-operations">Operationen mit
Funktionen</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s04.html">Trenner</a></span></dt><dt><span class="sect1"><a
href="ch05s05.html">Kommentare</a></span></dt><dt><span class="sect1"><a href="ch05s06.html">Modulare
Auswertung</a></span></dt><dt><span class="sect1"><a href="ch05s07.html">Liste der
GEL-Operatoren</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch06.html">6. Programmierung mit
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch06.html#genius-gel-conditionals">Bedingungen</a></span></dt><dt><span class="sect1"><a
href="ch06s02.html">Schleifen</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-while">While-Schleifen</a></span></dt><dt><span cla
ss="sect2"><a href="ch06s02.html#genius-gel-loops-for">For-Schleifen</a></span></dt><dt><span
class="sect2"><a href="ch06s02.html#genius-gel-loops-foreach">Foreach-Schleifen</a></span></dt><dt><span
class="sect2"><a href="ch06s02.html#genius-gel-loops-break-continue">Break and
Continue</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch06s03.html">Summen und
Produkte</a></span></dt><dt><span class="sect1"><a
href="ch06s04.html">Vergleichsoperatoren</a></span></dt><dt><span class="sect1"><a
href="ch06s05.html">Globale Variablen und Variablenbereiche</a></span></dt><dt><span class="sect1"><a
href="ch06s06.html">Parametervariablen</a></span></dt><dt><span class="sect1"><a
href="ch06s07.html">Rückgabewerte</a></span></dt><dt><span class="sect1"><a
href="ch06s08.html">Referenzen</a></span></dt><dt><span class="sect1"><a href="ch06s09.html">Lvalues (linke
Werte)</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch07.html">7. Fortgeschrittene
Programmierung mit
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch07.html#genius-gel-error-handling">Fehlerbehandlung</a></span></dt><dt><span class="sect1"><a
href="ch07s02.html">Übergeordnete Syntax</a></span></dt><dt><span class="sect1"><a
href="ch07s03.html">Funktionen als Rückgabe</a></span></dt><dt><span class="sect1"><a
href="ch07s04.html">Echte lokale Variablen</a></span></dt><dt><span class="sect1"><a href="ch07s05.html">GEL
Startprozedur</a></span></dt><dt><span class="sect1"><a href="ch07s06.html">Laden von
Programmen</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch08.html">8. Matrizen in
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch08.html#genius-gel-matrix-support">Matrizen
eingeben</a></span></dt><dt><span class="sect1"><a href="ch08s02.html">Operatoren für konjugierte
Transposition und Transposition</a></span></dt><dt><span class="sect1"><a href="ch08s03.html">Lineare
Algebra</a></span></dt></dl></dd><dt><span class="chapter"><a h
ref="ch09.html">9. Polynome in GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch09.html#genius-gel-polynomials-using">Verwendung von Polynomen</a></span></dt></dl></dd><dt><span
class="chapter"><a href="ch10.html">10. Mengenlehre in GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch10.html#genius-gel-sets-using">Mengen verwenden</a></span></dt></dl></dd><dt><span class="chapter"><a
href="ch11.html">11. Liste der GEL-Funktionen</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch11.html#genius-gel-function-list-commands">Befehle</a></span></dt><dt><span class="sect1"><a
href="ch11s02.html">Grundlegendes</a></span></dt><dt><span class="sect1"><a
href="ch11s03.html">Parameter</a></span></dt><dt><span class="sect1"><a
href="ch11s04.html">Konstanten</a></span></dt><dt><span class="sect1"><a
href="ch11s05.html">Numerik</a></span></dt><dt><span class="sect1"><a
href="ch11s06.html">Trigonometrie</a></span></dt><dt><span class="sect1"><a href="ch11s07.h
tml">Zahlentheorie</a></span></dt><dt><span class="sect1"><a
href="ch11s08.html">Matrixoperationen</a></span></dt><dt><span class="sect1"><a href="ch11s09.html">Lineare
Algebra</a></span></dt><dt><span class="sect1"><a href="ch11s10.html">Kombinatorik</a></span></dt><dt><span
class="sect1"><a href="ch11s11.html">Analysis</a></span></dt><dt><span class="sect1"><a
href="ch11s12.html">Funktionen</a></span></dt><dt><span class="sect1"><a href="ch11s13.html">Gleichungen
lösen</a></span></dt><dt><span class="sect1"><a href="ch11s14.html">Statistik</a></span></dt><dt><span
class="sect1"><a href="ch11s15.html">Polynomials</a></span></dt><dt><span class="sect1"><a
href="ch11s16.html">Mengenlehre</a></span></dt><dt><span class="sect1"><a href="ch11s17.html">Commutative
Algebra</a></span></dt><dt><span class="sect1"><a href="ch11s18.html">Verschiedenes</a></span></dt><dt><span
class="sect1"><a href="ch11s19.html">Symbolische Operationen</a></span></dt><dt><span class="sect1"><a href="
ch11s20.html">Darstellung</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch12.html">12.
Beispielprogramme in GEL</a></span></dt><dt><span class="chapter"><a href="ch13.html">13.
Einstellungen</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch13.html#genius-prefs-output">Ausgabe</a></span></dt><dt><span class="sect1"><a
href="ch13s02.html">Genauigkeit</a></span></dt><dt><span class="sect1"><a
href="ch13s03.html">Terminal</a></span></dt><dt><span class="sect1"><a
href="ch13s04.html">Speicher</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch14.html">14. Info
zu <span class="application">Genius Mathematikwerkzeug</span></a></span></dt></dl></div><div
class="list-of-figures"><p><b>Abbildungsverzeichnis</b></p><dl><dt>2.1. <a
href="ch02s02.html#mainwindow-fig"><span class="application">Genius
Mathematikwerkzeug</span>-Fenster</a></dt><dt>4.1. <a href="ch04.html#lineplot-fig">Fenster »Darstellung
erstellen«</a></dt><dt>4.2. <a href="ch04.html#lin
eplot2-fig">Fenster »Darstellen«</a></dt><dt>4.3. <a href="ch04s02.html#paramplot-fig">Reiter »Parametrische
Darstellung«</a></dt><dt>4.4. <a href="ch04s02.html#paramplot2-fig">Parametrische
Darstellung</a></dt><dt>4.5. <a
href="ch04s05.html#surfaceplot-fig">2D-Darstellung</a></dt></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"> </td><td width="20%"
align="center"> </td><td width="40%" align="right"> <a accesskey="n"
href="ch01.html">Weiter</a></td></tr><tr><td width="40%" align="left" valign="top"> </td><td width="20%"
align="center"> </td><td width="40%" align="right" valign="top"> Kapitel 1.
Einführung</td></tr></table></div></body></html>
diff --git a/help/el/html/ch05s07.html b/help/el/html/ch05s07.html
index 69df3bd..98d65fc 100644
--- a/help/el/html/ch05s07.html
+++ b/help/el/html/ch05s07.html
@@ -11,15 +11,57 @@ returns 3.
</p><p>Δείτε <a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html";
target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Εκθετοποίηση, ανυψώνει μια <code
class="varname">a</code> στη δύναμη <code class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Εκθετοποίηση στοιχείου κατά στοιχείο. Ανυψώνει
κάθε στοιχείο ενός πίνακα <code class="varname">a</code> στη δύναμη <code class="varname">b</code>. Ή αν η
<code class="varname">b</code> είναι ένας πίνακας του ίδιου μεγέθους όπως η <code class="varname">a</code>,
τότε κάνει την πράξη στοιχείο κατά στοιχείο. Αν η <code class
="varname">a</code> είναι ένας αριθμός και η <code class="varname">b</code> είναι ένας πίνακας, τότε
δημιουργεί έναν πίνακα του ίδιου μεγέθους όπως η <code class="varname">b</code> με τη <code
class="varname">a</code> υψωμένη σε όλες τις διαφορετικές δυνάμεις στην <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a+b</code></strong></span></dt><dd><p>Πρόσθεση. Προσθέτει δύο αριθμούς, πίνακες,
συναρτήσεις ή συμβολοσειρές. Αν προσθέτετε μια συμβολοσειρά σε ο,τιδήποτε το αποτέλεσμα θα είναι απλά μια
συμβολοσειρά. Αν ο ένας είναι ένας τετραγωνικός πίνακας και ο άλλος ένας αριθμός, τότε ο αριθμός πολλαπλα�
�ιάζεται με τον ταυτοτικό πίνακα.</p></dd><dt><span class="term"><strong
class="userinput"><code>a-b</code></strong></span></dt><dd><p>Αφαίρεση. Αφαιρεί δύο αριθμούς, πίνακες ή
συναρτήσεις.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Πολλαπλασιασμός. Αυτός είναι ο κανονικός
πίνακας πολλαπλασιασμού.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Πολλαπλασιασμός στοιχείο με στοιχείο αν οι
<code class="varname">a</code> και <code class="varname">b</code> είναι πίνακες.</p></dd><dt><span
class="term"><strong class="userinput"><code>a/b</code></strong></span></dt><dd><p>Διαίρεση. Όταν οι <code
class="varname">a</code> και <code class="varname">b</code> είναι μόνο α�
�ιθμοί, αυτή είναι η κανονική διαίρεση. Όταν είναι πίνακες, τότε αυτή είναι ισοδύναμη με <strong
class="userinput"><code>a*b^-1</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>
Element by element division. Same as <strong class="userinput"><code>a/b</code></strong>
for
numbers, but operates element by element on matrices.
- </p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Οπίσθια διαίρεση. Είναι η ίδια με <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Οπίσθια διαίρεση στοιχείου με
στοιχείο.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>Ο τελεστής mod (ισοϋπόλοιπο). Αυτός δεν
ενεργοποιεί την <a class="link" href="ch05s06.html" title="Μετρικός υπολογισμός">κατάσταση υπολοίπων</a>,
αλλά επιστρέφει απλά το υπόλοιπο της <strong class="userinput"><code>a/b</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.%b</code></strong></span></dt><dd><p>Ο τελεστής ισοϋπόλο�
�που στοιχείου κατά στοιχείο. Επιστρέφει το ακέραιο υπόλοιπο μετά το στοιχείο κατά στοιχείο του <strong
class="userinput"><code>a./b</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a mod b</code></strong></span></dt><dd><p>Τελεστής υπολογισμού υπολοίπων. Η παράσταση
<code class="varname">a</code> υπολογίζει το modulo <code class="varname">b</code>. Δείτε <a class="xref"
href="ch05s06.html" title="Μετρικός υπολογισμός">«Μετρικός υπολογισμός»</a>. Κάποιες συναρτήσεις και κάποιοι
τελεστές συμπεριφέρονται διαφορετικά με το ισοϋπόλοιπο ενός ακεραίου.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Παραγοντικός τελεστής
. Αυτό είναι παρόμοιο με <strong
class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Διπλός παραγοντικός τελεστής. Αυτός είναι
παρόμοιος με <strong class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a==b</code></strong></span></dt><dd><p>Τελεστής ισότητας.
Επιστρέφει <code class="constant">αληθές</code> ή <code class="constant">ψευδές</code> ανάλογα με το αν οι
<code class="varname">a</code> και <code class="varname">b</code> είναι ίσες ή όχι.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Τελεστής ανισότητας,
επιστρέφει <code class="constant">αληθές</code> αν η <code cla
ss="varname">a</code> δεν είναι ίση με την <code class="varname">b</code>, αλλιώς επιστρέφει <code
class="constant">ψευδές</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Εναλλακτικός τελεστής ανισότητας,
επιστρέφει <code class="constant">αληθές</code> αν η <code class="varname">a</code> δεν είναι ίση με την
<code class="varname">b</code>, αλλιώς επιστρέφει <code class="constant">ψευδές</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Τελεστής μικρότερος
από ή ίσος, επιστρέφει <code class="constant">αληθές</code> αν <code class="varname">a</code> είναι μικρότερο
από ή ίσο με <code class="varname">b</code>, αλλιώς επιστρέφει <code class="cons
tant">ψευδές</code>. Αυτοί μπορούν να συνδεθούν όπως στο <strong class="userinput"><code>a <= b <=
c</code></strong> (μπορούν επίσης να συνδυαστούν με τον τελεστή λιγότερο από).</p></dd><dt><span
class="term"><strong class="userinput"><code>a>=b</code></strong></span></dt><dd><p>Τελεστής μεγαλύτερος
από ή ίσος, επιστρέφει <code class="constant">αληθές</code> αν η <code class="varname">a</code> είναι
μεγαλύτερη από ή ίση με την <code class="varname">b</code>, αλλιώς επιστρέφει <code
class="constant">ψευδές</code>. Αυτοί μπορούν να συνδεθούν όπως στο <strong class="userinput"><code>a >= b
>= c</code></strong> (μπορούν επίσης να συνδυαστούν με τον τελεστή μεγαλύτερο από).</p></dd><dt><span
class="term"><str
ong class="userinput"><code>a<b</code></strong></span></dt><dd><p>Τελεστής μικρότερος από, επιστρέφει
<code class="constant">αληθές</code> αν <code class="varname">a</code> είναι μικρότερη από <code
class="varname">b</code>, αλλιώς επιστρέφει <code class="constant">ψευδές</code>. Αυτοί μπορούν να συνδεθούν
όπως στο <strong class="userinput"><code>a < b < c</code></strong> (μπορούν επίσης να συνδυαστούν με
τον τελεστή μικρότερο από ή ίσο).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>Τελεστής μεγαλύτερος από, επιστρέφει <code
class="constant">αληθές</code> αν η <code class="varname">a</code> είναι μικρότερη από <code
class="varname">b</code>, αλλιώς επιστρέφει <code class="cons
tant">ψευδές</code>. Αυτοί μπορούν να συνδεθούν όπως στο <strong class="userinput"><code>a > b >
c</code></strong> (μπορούν επίσης να συνδυαστούν με τον τελεστή μεγαλύτερο από ή ίσο).</p></dd><dt><span
class="term"><strong class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Τελεστής
σύγκρισης. Αν <code class="varname">a</code> είναι ίσο με <code class="varname">b</code> επιστρέφει 0, αν
<code class="varname">a</code> είναι μικρότερο από <code class="varname">b</code> επιστρέφει -1 και αν <code
class="varname">a</code> είναι μεγαλύτερο από <code class="varname">b</code> επιστρέφει 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a και b</code></strong></span></dt><dd><p>Λογικό και. Επιστρέφει
αληθές αν αμφότερ�
� τα <code class="varname">a</code> και <code class="varname">b</code> είναι αληθή, αλλιώς επιστρέφει
ψευδές. Αν είναι δοσμένοι οι αριθμοί, οι μη μηδενικοί αριθμοί αντιμετωπίζονται ως αληθείς.</p></dd><dt><span
class="term"><strong class="userinput"><code>a ή b</code></strong></span></dt><dd><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Οπίσθια διαίρεση. Είναι η ίδια με <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Οπίσθια διαίρεση στοιχείου με
στοιχείο.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>
+ The mod operator. This does not turn on the <a class="link" href="ch05s06.html" title="Μετρικός
υπολογισμός">modular mode</a>, but
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>Τελεστής υπολογισμού υπολοίπων. Η παράσταση <code
class="varname">a</code> υπολογίζει το modulo <code class="varname">b</code>. Δείτε <a class="xref"
href="ch05s06.html" title="Μετρικός υπολογισμός">«Μετρικός υπολογισμός»</a>. Κάποιες συναρτήσεις και κάποιοι
τελεστές συμπεριφέρονται διαφορετικά με το ισοϋπόλοιπο ενός ακεραίου.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Παραγοντικός τελεστής. Αυτό είναι παρόμοιο με
<strong class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Δι
πλός παραγοντικός τελεστής. Αυτός είναι παρόμοιος με <strong
class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p>Τελεστής ισότητας. Επιστρέφει <code
class="constant">αληθές</code> ή <code class="constant">ψευδές</code> ανάλογα με το αν οι <code
class="varname">a</code> και <code class="varname">b</code> είναι ίσες ή όχι.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Τελεστής ανισότητας,
επιστρέφει <code class="constant">αληθές</code> αν η <code class="varname">a</code> δεν είναι ίση με την
<code class="varname">b</code>, αλλιώς επιστρέφει <code class="constant">ψευδές</code>.</p></dd><dt><span
class="term"><strong class="use
rinput"><code>a<>b</code></strong></span></dt><dd><p>Εναλλακτικός τελεστής ανισότητας, επιστρέφει
<code class="constant">αληθές</code> αν η <code class="varname">a</code> δεν είναι ίση με την <code
class="varname">b</code>, αλλιώς επιστρέφει <code class="constant">ψευδές</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Τελεστής μικρότερος
από ή ίσος, επιστρέφει <code class="constant">αληθές</code> αν <code class="varname">a</code> είναι μικρότερο
από ή ίσο με <code class="varname">b</code>, αλλιώς επιστρέφει <code class="constant">ψευδές</code>. Αυτοί
μπορούν να συνδεθούν όπως στο <strong class="userinput"><code>a <= b <= c</code></strong> (μπορούν
επίσης να συνδυαστούν μ
ε τον τελεστή λιγότερο από).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>
+ Greater than or equal operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than or equal to
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
+ (and they can also be combined with the greater than operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
+ Less than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ less than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
+ (they can also be combined with the less than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
+ Greater than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
+ (they can also be combined with the greater than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Τελεστής σύγκρισης. Αν <code
class="varname">a</code> είναι ίσο με <code class="varname">b</code> επιστρέφει 0, αν <code
class="varname">a</code> είναι μικρότερο από <code class="varname">b</code> επιστρέφει -1 και αν <code
class="varname">a</code> είναι μεγαλύτερο από <code class="varname">b</code> επιστρέφει 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a και b</code></strong></span></dt><dd><p>Λογικό και. Επιστρέφει
αληθές αν αμφότερα τα <code class="varname">a</code> και <code class="varname">b</code> είναι αληθή, αλλιώς
επιστρέφει ψευδές. Αν είναι δοσμένοι οι αριθμοί, οι μη μηδενικοί αριθμοί αντιμετωπίζο
νται ως αληθείς.</p></dd><dt><span class="term"><strong class="userinput"><code>a ή
b</code></strong></span></dt><dd><p>
Logical or.
Returns true if either
<code class="varname">a</code> or <code class="varname">b</code> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
- </p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>Λογικό αποκλειστικό ή (xor). Επιστρέφει αληθές αν ένα ακριβώς από τα
<code class="varname">a</code> ή <code class="varname">b</code> είναι αληθές, αλλιώς επιστρέφει ψευδές. Αν οι
αριθμοί είναι δοσμένοι, οι μη μηδενικοί αριθμοί αντιμετωπίζονται ως αληθείς.</p></dd><dt><span
class="term"><strong class="userinput"><code>όχι a</code></strong></span></dt><dd><p>Λογικό όχι. Επιστρέφει
την λογική άρνηση του <code class="varname">a</code></p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
+ Logical xor.
+ Returns true if exactly one of
+ <code class="varname">a</code> or <code class="varname">b</code> is true,
+ else returns false. If given numbers, nonzero numbers
+ are treated as true.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>όχι
a</code></strong></span></dt><dd><p>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
- </p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Αναφορά μεταβλητής (για το πέρασμα μιας
αναφοράς σε μια μεταβλητή). Δείτε <a class="xref" href="ch06s08.html"
title="Αναφορές">«Αναφορές»</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>Αποαναφορά μεταβλητής (για πρόσβαση σε μια
αναφερθείσα μεταβλητή). Δείτε <a class="xref" href="ch06s08.html"
title="Αναφορές">«Αναφορές»</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Αναστροφή συζυγούς πίνακα. Δηλαδή, οι γραμμές
και οι στήλες εναλλάσσονται και παίρνουμε τον συζυγή μιγαδικό όλων των καταχωρί
σεων. Δηλαδή αν τα στοιχεία i,j της <code class="varname">a</code> είναι x+iy, τότε τα στοιχεία j,i του
<strong class="userinput"><code>a'</code></strong> είναι x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Η αναστροφή πίνακα, δεν παίρνει τον συζυγή
μιγαδικό των καταχωρίσεων. Δηλαδή, τα στοιχεία i,j της <code class="varname">a</code> γίνονται τα στοιχεία
του <strong class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>Λήψη στοιχείου ενός πίνακα στη γραμμή <code
class="varname">b</code> και στήλη <code class="varname">c</code>. Αν οι <code class="varname">b</code>,
<code class="varname">c</code> είναι διανύσματα, τ
ότε αυτό παίρνει τις αντίστοιχες στήλες γραμμές ή υποπίνακες.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Λήψη γραμμής ενός πίνακα (ή πολλαπλών
γραμμών αν το <code class="varname">b</code> είναι ένα διάνυσμα).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Ίδιο με το παραπάνω.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Λήψη στήλης ενός πίνακα
(ή στηλών αν το <code class="varname">c</code> είναι ένα διάνυσμα).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Ίδιο με το παραπάνω.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(b)</code></stron
g></span></dt><dd><p>Λήψη ενός στοιχείου από έναν πίνακα αντιμετωπίζοντας τον ως διάνυσμα. Αυτό θα διατρέξει
τον πίνακα κατά τη γραμμή.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Δόμηση ενός διανύσματος από το <code
class="varname">a</code> στο <code class="varname">b</code> (ή ορίστε γραμμή, περιοχή στήλης για τον τελεστή
<code class="literal">@</code>). Για παράδειγμα για να πάρετε τις γραμμές 2 μέχρι 4 του πίνακα <code
class="varname">Α</code> μπορούμε να κάνουμε </p><pre class="programlisting">A@(2:4,)
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Αναφορά μεταβλητής (για το πέρασμα μιας
αναφοράς σε μια μεταβλητή). Δείτε <a class="xref" href="ch06s08.html"
title="Αναφορές">«Αναφορές»</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>Αποαναφορά μεταβλητής (για πρόσβαση σε μια
αναφερθείσα μεταβλητή). Δείτε <a class="xref" href="ch06s08.html"
title="Αναφορές">«Αναφορές»</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Αναστροφή συζυγούς πίνακα. Δηλαδή, οι γραμμές
και οι στήλες εναλλάσσονται και παίρνουμε τον συζυγή μιγαδικό όλων των καταχωρί
σεων. Δηλαδή αν τα στοιχεία i,j της <code class="varname">a</code> είναι x+iy, τότε τα στοιχεία j,i του
<strong class="userinput"><code>a'</code></strong> είναι x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Η αναστροφή πίνακα, δεν παίρνει τον συζυγή
μιγαδικό των καταχωρίσεων. Δηλαδή, τα στοιχεία i,j της <code class="varname">a</code> γίνονται τα στοιχεία
του <strong class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>
+ Get element of a matrix in row <code class="varname">b</code> and column
+ <code class="varname">c</code>. If <code class="varname">b</code>,
+ <code class="varname">c</code> are vectors, then this gets the corresponding
+ rows, columns or submatrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Λήψη γραμμής ενός πίνακα (ή πολλαπλών
γραμμών αν το <code class="varname">b</code> είναι ένα διάνυσμα).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Ίδιο με το παραπάνω.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Λήψη στήλης ενός πίνακα
(ή στηλών αν το <code class="varname">c</code> είναι ένα διάνυσμα).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Ίδιο με το παραπάνω.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Λήψη ενός στοιχείου από
έναν πίνακα αντιμετ�
�πίζοντας τον ως διάνυσμα. Αυτό θα διατρέξει τον πίνακα κατά τη γραμμή.</p></dd><dt><span
class="term"><strong class="userinput"><code>a:b</code></strong></span></dt><dd><p>Δόμηση ενός διανύσματος
από το <code class="varname">a</code> στο <code class="varname">b</code> (ή ορίστε γραμμή, περιοχή στήλης για
τον τελεστή <code class="literal">@</code>). Για παράδειγμα για να πάρετε τις γραμμές 2 μέχρι 4 του πίνακα
<code class="varname">Α</code> μπορούμε να κάνουμε </p><pre class="programlisting">A@(2:4,)
</pre><p> ως <strong class="userinput"><code>2:4</code></strong> που θα επιστρέψει ένα διάνυσμα
<strong class="userinput"><code>[2,3,4]</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b:c</code></strong></span></dt><dd><p>Δόμηση ενός διανύσματος από <code
class="varname">a</code> σε <code class="varname">c</code> με <code class="varname">b</code> ως ένα βήμα.
Δηλαδή για παράδειγμα </p><pre class="programlisting">genius> 1:2:9
=
`[1, 3, 5, 7, 9]
@@ -43,5 +85,10 @@ returns 3.
<strong class="userinput"><code>float(1:2/5:3)</code></strong> even gives you floating
point numbers and is ever so slightly more precise than
<strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
- </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Κάντε έναν φανταστικό αριθμό (πολλαπλασιάστε
το <code class="varname">a</code> με τον φανταστικό). Σημειώστε ότι, κανονικά ο αριθμός <code
class="varname">i</code> γράφεται ως <strong class="userinput"><code>1i</code></strong>. Έτσι το παραπάνω
είναι ίσο με </p><pre class="programlisting">(a)*1i
- </pre></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Βάλτε ` σε ένα αναγνωριστικό έτσι ώστε να μην
υπολογιστεί. Ή βάλτε ` σε έναν πίνακα, έτσι ώστε να μην επεκταθεί.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Εναλλαγή τιμής του <code
class="varname">a</code> με την τιμή του <code class="varname">b</code>. Πρός το παρόν δεν λειτουργεί σε
περιοχές στοιχείων πίνακα. Επιστρέφει <code class="constant">null</code>. Διαθέσιμο από την έκδοση
1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment
a</code></strong></span></dt><dd><p>Αύξηση της μεταβλητής <code class="varname">a</code> κατά 1. Αν η <code
class="varname">a<
/code> είναι ένας πίνακας, τότε αυξάνεται κάθε στοιχείο. Αυτό είναι ισοδύναμο με το <strong
class="userinput"><code>a=a+1</code></strong>, αλλά είναι κάπως γρηγορότερο. Επιστρέφει <code
class="constant">null</code>. Διαθέσιμο από την έκδοση 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a by b</code></strong></span></dt><dd><p>Αυξάνει τη μεταβλητή <code
class="varname">a</code> κατά <code class="varname">b</code>. Αν η <code class="varname">a</code> είναι ένας
πίνακας, τότε αυξάνεται κάθε στοιχείο. Αυτό είναι ισοδύναμο με το <strong
class="userinput"><code>a=a+b</code></strong>, αλλά είναι κάπως γρηγορότερο. Επιστρέφει <code
class="constant">null</code>. Διαθέσιμο από την έκδοση 1.0.
13.</p></dd></dl></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Σημείωση</h3><p>Ο τελεστής @() καθιστά τον: τελεστή πιο χρήσιμο. Με αυτό μπορείτε να ορίσετε
περιοχές ενός πίνακα. Έτσι ώστε a@(2:4,6) είναι οι γραμμές 2,3,4 της στήλης 6. Ή a@(,1:2) θα σας πάρει τις
πρώτες δύο στήλες ενός πίνακα. Μπορείτε επίσης να αναθέσετε στον τελεστή @(), όσο η δεξιά τιμή είναι ο
πίνακας που ταιριάζει στην περιοχή σε μέγεθος, ή αν είναι οποιουδήποτε άλλου τύπου τιμής.</p></div><div
class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Σημείωση</h3><p>Οι τελεστές
σύγκρισης (εκτός από τον τελεστή <
=> που συμπεριφέρεται κανονικά), δεν είναι αυστηρά δυαδικοί τελεστές, μπορούν στην πραγματικότητα να
ομαδοποιηθούν με τον κανονικό μαθηματικό τρόπο, π.χ.: (1<x<=y<5) είναι μια επιτρεπτή παράσταση
λογικών τιμών και σημαίνει απλά αυτό που πρέπει, δηλαδή (1<x and x≤y and y<5)</p></div><div
class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Σημείωση</h3><p>Ο μοναδιαίος
τελεστής μείον λειτουργεί με διαφορετικό τρόπο ανάλογα με το πού εμφανίζεται. Αν εμφανίζεται πριν από ένα
αριθμό έχει στενή προτεραιότητα, αν εμφανίζεται μπροστά από μια παράσταση έχει μικρότε
ρη προτεραιότητα από τη δύναμη και τους παραγοντικούς τελεστές. Έτσι για παράδειγμα <strong
class="userinput"><code>-1^k</code></strong> είναι στην πραγματικότητα <strong
class="userinput"><code>(-1)^k</code></strong>, αλλά <strong
class="userinput"><code>-foo(1)^k</code></strong> είναι στην πραγματικότητα <strong
class="userinput"><code>-(foo(1)^k)</code></strong>. Γιαυτό να προσέχετε τη χρήση του και αν αμφιβάλετε,
προσθέστε παρενθέσεις.</p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch05s06.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch05.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Επόμενο</a></td></tr><tr><
td width="40%" align="left" valign="top">Μετρικός υπολογισμός </td><td width="20%" align="center"><a
accesskey="h" href="index.html">Αρχή</a></td><td width="40%" align="right" valign="top"> Κεφάλαιο 6.
Προγραμματισμός με GEL</td></tr></table></div></body></html>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
+ written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
+ </p><pre class="programlisting">(a)*1i
+ </pre><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Βάλτε ` σε ένα αναγνωριστικό έτσι ώστε να μην
υπολογιστεί. Ή βάλτε ` σε έναν πίνακα, έτσι ώστε να μην επεκταθεί.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Εναλλαγή τιμής του <code
class="varname">a</code> με την τιμή του <code class="varname">b</code>. Πρός το παρόν δεν λειτουργεί σε
περιοχές στοιχείων πίνακα. Επιστρέφει <code class="constant">null</code>. Διαθέσιμο από την έκδοση
1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment
a</code></strong></span></dt><dd><p>Αύξηση της μεταβλητής <code class="varname">a</code> κατά 1. Αν η <code
class="varname"
a</code> είναι ένας πίνακας, τότε αυξάνεται κάθε στοιχείο. Αυτό είναι ισοδύναμο με το <strong
class="userinput"><code>a=a+1</code></strong>, αλλά είναι κάπως γρηγορότερο. Επιστρέφει <code
class="constant">null</code>. Διαθέσιμο από την έκδοση 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a by b</code></strong></span></dt><dd><p>Αυξάνει τη μεταβλητή <code
class="varname">a</code> κατά <code class="varname">b</code>. Αν η <code class="varname">a</code> είναι
ένας πίνακας, τότε αυξάνεται κάθε στοιχείο. Αυτό είναι ισοδύναμο με το <strong
class="userinput"><code>a=a+b</code></strong>, αλλά είναι κάπως γρηγορότερο. Επιστρέφει <code
class="constant">null</code>. Διαθέσιμο από την έκδοση 1
.0.13.</p></dd></dl></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Σημείωση</h3><p>Ο τελεστής @() καθιστά τον: τελεστή πιο χρήσιμο. Με αυτό μπορείτε να ορίσετε
περιοχές ενός πίνακα. Έτσι ώστε a@(2:4,6) είναι οι γραμμές 2,3,4 της στήλης 6. Ή a@(,1:2) θα σας πάρει τις
πρώτες δύο στήλες ενός πίνακα. Μπορείτε επίσης να αναθέσετε στον τελεστή @(), όσο η δεξιά τιμή είναι ο
πίνακας που ταιριάζει στην περιοχή σε μέγεθος, ή αν είναι οποιουδήποτε άλλου τύπου τιμής.</p></div><div
class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Σημείωση</h3><p>Οι τελεστές
σύγκρισης (εκτός από τον τελεστή &
lt;=> που συμπεριφέρεται κανονικά), δεν είναι αυστηρά δυαδικοί τελεστές, μπορούν στην πραγματικότητα να
ομαδοποιηθούν με τον κανονικό μαθηματικό τρόπο, π.χ.: (1<x<=y<5) είναι μια επιτρεπτή παράσταση
λογικών τιμών και σημαίνει απλά αυτό που πρέπει, δηλαδή (1<x and x≤y and y<5)</p></div><div
class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Σημείωση</h3><p>Ο μοναδιαίος
τελεστής μείον λειτουργεί με διαφορετικό τρόπο ανάλογα με το πού εμφανίζεται. Αν εμφανίζεται πριν από ένα
αριθμό έχει στενή προτεραιότητα, αν εμφανίζεται μπροστά από μια παράσταση έχει μικρό�
�ερη προτεραιότητα από τη δύναμη και τους παραγοντικούς τελεστές. Έτσι για παράδειγμα <strong
class="userinput"><code>-1^k</code></strong> είναι στην πραγματικότητα <strong
class="userinput"><code>(-1)^k</code></strong>, αλλά <strong
class="userinput"><code>-foo(1)^k</code></strong> είναι στην πραγματικότητα <strong
class="userinput"><code>-(foo(1)^k)</code></strong>. Γιαυτό να προσέχετε τη χρήση του και αν αμφιβάλετε,
προσθέστε παρενθέσεις.</p></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch05s06.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch05.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Επόμενο</a></td></tr><t
r><td width="40%" align="left" valign="top">Μετρικός υπολογισμός </td><td width="20%" align="center"><a
accesskey="h" href="index.html">Αρχή</a></td><td width="40%" align="right" valign="top"> Κεφάλαιο 6.
Προγραμματισμός με GEL</td></tr></table></div></body></html>
diff --git a/help/el/html/ch06s05.html b/help/el/html/ch06s05.html
index c49d058..41ad02f 100644
--- a/help/el/html/ch06s05.html
+++ b/help/el/html/ch06s05.html
@@ -1,4 +1,12 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Καθολικές μεταβλητές
και εμβέλεια μεταβλητών</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch06.html" title="Κεφάλαιο 6.
Προγραμματισμός με GEL"><link rel="prev" href="ch06s04.html" title="Τελεστές σύγκρισης"><link rel="next"
href="ch06s06.html" title="Μεταβλητές παραμέτρων"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Καθολικές μεταβλητές και εμβέλεια μεταβλητών</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Προηγ</a> </td><th width="60%" align="cent
er">Κεφάλαιο 6. Προγραμματισμός με GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Καθολικές μεταβλητές και εμβέλεια
μεταβλητών</h2></div></div></div><p>Η GEL είναι μια <a class="ulink"
href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">δυναμική γλώσσα με εμβέλεια</a>. Θα
εξηγήσουμε τι σημαίνει αυτό παρακάτω. Δηλαδή, κανονικές μεταβλητές και συναρτήσεις είναι δυναμικά με
εμβέλεια. Η εξαίρεση είναι οι <a class="link" href="ch06s06.html" title="Μεταβλητές παραμέτρων">μεταβλητές
παραμέτρου</a>, που εί�
�αι πάντα καθολικές.</p><p>Όπως οι περισσότερες γλώσσες προγραμματισμού, η GEL έχει διαφορετικούς τύπους
μεταβλητών. Κανονικά, όταν μια μεταβλητή ορίζεται σε μια συνάρτηση, είναι ορατή από αυτή τη συνάρτηση και από
όλες τις συναρτήσεις που καλούνται (όλες με υψηλότερα περιεχόμενα). Για παράδειγμα, ας υποθέσουμε ότι μια
συνάρτηση <code class="function">f</code> ορίζει μια μεταβλητή <code class="varname">a</code> και έπειτα
καλεί τη συνάρτηση <code class="function">g</code>. Τότε η συνάρτηση <code class="function">g</code> μπορεί
να αναφέρει την <code class="varname">a</code>. Αλλά μόλις η <code class="function">f</code> επιστ
ρέφει, η μεταβλητή <code class="varname">a</code> βγαίνει εκτός εμβέλειας. Για παράδειγμα, ο παρακάτω
κώδικας θα εμφανίσει 5. Η συνάρτηση <code class="function">g</code> δεν μπορεί να κληθεί στο ανώτατο επίπεδο
(έξω από τη <code class="function">f</code> ως <code class="varname">a</code> δεν θα οριστεί). </p><pre
class="programlisting">function f() = (a:=5; g());
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Καθολικές μεταβλητές
και εμβέλεια μεταβλητών</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch06.html" title="Κεφάλαιο 6.
Προγραμματισμός με GEL"><link rel="prev" href="ch06s04.html" title="Τελεστές σύγκρισης"><link rel="next"
href="ch06s06.html" title="Μεταβλητές παραμέτρων"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Καθολικές μεταβλητές και εμβέλεια μεταβλητών</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Προηγ</a> </td><th width="60%" align="cent
er">Κεφάλαιο 6. Προγραμματισμός με GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Καθολικές μεταβλητές και εμβέλεια μεταβλητών</h2></div></div></div><p>
+ GEL is a
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ dynamically scoped language</a>. We will explain what this
+ means below. That is, normal variables and functions are dynamically
+ scoped. The exception are
+ <a class="link" href="ch06s06.html" title="Μεταβλητές παραμέτρων">parameter variables</a>,
+ which are always global.
+ </p><p>Όπως οι περισσότερες γλώσσες προγραμματισμού, η GEL έχει διαφορετικούς τύπους μεταβλητών.
Κανονικά, όταν μια μεταβλητή ορίζεται σε μια συνάρτηση, είναι ορατή από αυτή τη συνάρτηση και από όλες τις
συναρτήσεις που καλούνται (όλες με υψηλότερα περιεχόμενα). Για παράδειγμα, ας υποθέσουμε ότι μια συνάρτηση
<code class="function">f</code> ορίζει μια μεταβλητή <code class="varname">a</code> και έπειτα καλεί τη
συνάρτηση <code class="function">g</code>. Τότε η συνάρτηση <code class="function">g</code> μπορεί να
αναφέρει την <code class="varname">a</code>. Αλλά μόλις η <code class="function">f</code> επιστρέφει, η
μεταβλητή <
code class="varname">a</code> βγαίνει εκτός εμβέλειας. Για παράδειγμα, ο παρακάτω κώδικας θα εμφανίσει 5. Η
συνάρτηση <code class="function">g</code> δεν μπορεί να κληθεί στο ανώτατο επίπεδο (έξω από τη <code
class="function">f</code> ως <code class="varname">a</code> δεν θα οριστεί). </p><pre
class="programlisting">function f() = (a:=5; g());
function g() = print(a);
f();
</pre><p>Αν ορίσετε μια μεταβλητή μέσα σε μια συνάρτηση θα επικαλύψει οποιεσδήποτε μεταβλητές ορίστηκαν στις
συναρτήσεις κλήσης. Για παράδειγμα, τροποποιούμε τον παραπάνω κώδικα και γράφουμε: </p><pre
class="programlisting">function f() = (a:=5; g());
diff --git a/help/el/html/ch07s02.html b/help/el/html/ch07s02.html
index ebd7cd1..0c30546 100644
--- a/help/el/html/ch07s02.html
+++ b/help/el/html/ch07s02.html
@@ -3,10 +3,32 @@
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
- as just sequence of lines as if were entered on the command line.
+ as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
- parentheses).
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
</p><p>Ο παρακάτω κώδικας θα παράξει ένα σφάλμα όταν εισαχθεί στο ανώτατο επίπεδο ενός προγράμματος,
ενώ θα δουλέψει θαυμάσια σε μια συνάρτηση. </p><pre class="programlisting">if Something() then
DoSomething()
else
diff --git a/help/el/html/ch11s04.html b/help/el/html/ch11s04.html
index cbb78a5..356befc 100644
--- a/help/el/html/ch11s04.html
+++ b/help/el/html/ch11s04.html
@@ -1,23 +1,26 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Σταθερές</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html" title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της
GEL"><link rel="prev" href="ch11s03.html" title="Παράμετροι"><link rel="next" href="ch11s05.html"
title="Αριθμητικό"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Σταθερές</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s03.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος συναρτήσεων της
GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch11s05.html">Επό�
�ενο</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-function-list-constants"></a>Σταθερές</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Η σταθερά του Catalan, περίπου 0.915... Ορίζεται να είναι η σειρά,
όπου οι όροι είναι <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, με την <code
class="varname">k</code> να κυμαίνεται από 0 μέχρι το άπειρο.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Παραλλαγές: <code class="function">gamma</code></p><p>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>Η
χρυσή τομή.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
- round and uniform.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
+ round and uniform.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
is the exponential function
<a class="link" href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a>. It
is approximately
@@ -25,12 +28,12 @@
several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>Ο αριθμός π, που είναι ο λόγος της περιφέρειας ενός κύκλου προς τη διάμετρό του.
Αυτός είναι περίπου 3.14159265359...</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Παράμετροι </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td width="40%" align="right"
valign="top"> Αριθμητικό</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s05.html b/help/el/html/ch11s05.html
index 34e0c39..3acbd15 100644
--- a/help/el/html/ch11s05.html
+++ b/help/el/html/ch11s05.html
@@ -5,17 +5,35 @@
to <strong class="userinput"><code>|x|</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
<a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
<a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld
(complex modulus)</a>
for more information.
- </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Αντικατάσταση πολύ μικρού αριθμού με μηδέν.</p></dd><dt><span
class="term"><a name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Παραλλαγές: <code class="function">conj</code><code
class="function">Conj</code></p><p>Υπολογίζει τον μιγαδικό συζυγή του μιγαδικού αριθμού <code
class="varname">z</code>. Αν η <code class="varname">z</code> είναι ένα διάνυσμα ή πίνακας, όλα τα στοιχεία
του παίρνουν συζυγή.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-Denominator
"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator (x)</pre><p>Λήψη του παρανομαστή ενός
ρητού αριθμού.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Denominator";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Επιστροφή του κλασματικού μέρους ενός αριθμού.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Παραλλαγές: <code class="function">ImaginaryPart</code></p><p>Get the
imaginary part of a complex number. For example <strong class="
userinput"><code>Re(3+4i)</code></strong> yields 4.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Διαίρεση χωρίς υπόλοιπο.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
+ </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Αντικατάσταση πολύ μικρού αριθμού με μηδέν.</p></dd><dt><span
class="term"><a name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Παραλλαγές: <code class="function">conj</code><code
class="function">Conj</code></p><p>Υπολογίζει τον μιγαδικό συζυγή του μιγαδικού αριθμού <code
class="varname">z</code>. Αν η <code class="varname">z</code> είναι ένα διάνυσμα ή πίνακας, όλα τα στοιχεία
του παίρνουν συζυγή.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Λήψη του παρανομαστή ενός ρητού αριθμού.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Επιστροφή του κλασματικού μέρους ενός αριθμού.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Παραλλαγές: <code class="function">ImaginaryPart</code></p><p>Get the
imaginary part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong>
yields 4.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Διαίρεση χωρίς υπόλοιπο.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
<strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
<strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Ελέγχει αν το όρισμα είναι πιθανόν μιγαδικός ρητός αριθμός.
Δηλαδή, αν και το πραγματικό και το φανταστικό μέρος δίνονται ως ρητοί αριθμοί. Φυσικά ρητός σημαίνει απλά
"μη αποθηκευμένος ως αριθμός κινητής υποδιαστολής."</p></dd><dt><span class="term"><a
name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre class="synopsis">IsFloat (num)</pre><p>Check if
argument is a real floating point number (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(num)</p
re><p>Παραλλαγές: <code class="function">IsComplexInteger</code></p><p>Check if argument is a possibly
complex integer. That is a complex integer is a number of
the form <strong class="userinput"><code>n+1i*m</code></strong> where <code
class="varname">n</code> and <code class="varname">m</code>
- are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Ελέγχει αν το όρισμα είναι ένας ακέραιος (μη μιγαδικός).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Παραλλαγές: <code
class="function">IsNaturalNumber</code></p><p>Ελέγχει αν το όρισμα είναι θετικός πραγματικός ακέραιος.
Σημειώστε ότι δεχόμαστε τη σύμβαση ότι το 0 δεν είναι φυσικός αριθμ
ός.</p></dd><dt><span class="term"><a name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre
class="synopsis">IsRational (num)</pre><p>Ελέγχει αν το όρισμα είναι ένας ρητός αριθμός (μη μιγαδικός).
Φυσικά ρητός σημαίνει απλά "μη αποθηκευμένος ως αριθμός κινητής υποδιαστολής."</p></dd><dt><span
class="term"><a name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal
(num)</pre><p>Ελέγχει αν το όρισμα είναι ένας πραγματικός αριθμός.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Λήψη
του αριθμητή ενός ρητού αριθμού.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator";
target="_top">Wikipedia</a> για περισσότερες πληροφ
ορίες.</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Παραλλαγές: <code class="function">RealPart</code></p><p>Get the real part of
a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>Δείτε
<a class="ulink" href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Παραλλαγές: <code class="function">sign</code></p><p>Επιστρέφει το πρόσημο
ενός αριθμού. Δηλαδή επιστρέφει <code class="literal">-1</code> αν η τιμή είναι αρνητική, <code
class="literal">0</code> αν η τιμή είναι μηδέν και <code class="literal">1</code> αν η τιμή είναι θετική
. Αν <code class="varname">x</code> είναι μια μιγαδική τιμή τότε η <code class="function">Sign</code>
επιστρέφει την κατεύθυνση ή 0.</p></dd><dt><span class="term"><a
name="gel-function-ceil"></a>ceil</span></dt><dd><pre class="synopsis">ceil (x)</pre><p>Παραλλαγές: <code
class="function">Ceiling</code></p><p>Παίρνει τον πιο μικρό ακέραιο μεγαλύτερο από ή ίσο με τη <code
class="varname">n</code>. Παραδείγματα: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>ceil(1.1)</code></strong>
+ are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Ελέγχει αν το όρισμα είναι ένας ακέραιος (μη μιγαδικός).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Παραλλαγές: <code
class="function">IsNaturalNumber</code></p><p>Ελέγχει αν το όρισμα είναι θετικός πραγματικός ακέραιος.
Σημειώστε ότι δεχόμαστε τη σύμβαση ότι το 0 δεν είναι φυσικός αριθμ
ός.</p></dd><dt><span class="term"><a name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre
class="synopsis">IsRational (num)</pre><p>Ελέγχει αν το όρισμα είναι ένας ρητός αριθμός (μη μιγαδικός).
Φυσικά ρητός σημαίνει απλά "μη αποθηκευμένος ως αριθμός κινητής υποδιαστολής."</p></dd><dt><span
class="term"><a name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal
(num)</pre><p>Ελέγχει αν το όρισμα είναι ένας πραγματικός αριθμός.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Λήψη
του αριθμητή ενός ρητού αριθμού.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Παραλλαγές: <code class="function">RealPart</code></p><p>Get the real part of
a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Παραλλαγές: <code class="function">sign</code></p><p>Επιστρέφει το πρόσημο
ενός αριθμού. Δηλαδή επιστρέφει <code class="literal">-1</code> αν η τιμή είναι αρνητική, <code
class="literal">0</code> αν η τιμή είναι μηδέν και <code class="literal">1</code> αν η τιμή είναι θετική. Αν
<code class="varname">x</code> είναι μια μιγαδική τιμή τότε η <code class="function">Sign</code> επιστρέφει
την κατεύθυνση ή 0.</p></dd><dt><span class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre
class="synopsis">ceil (x)</pre><p>Παραλλαγές: <code class="function">Ceiling</code></p><p>Παίρνει τον πιο
μικρό ακέραιο μεγαλύτερο από ή ίσο μ
ε τη <code class="varname">n</code>. Παραδείγματα: </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>ceil(1.1)</code></strong>
= 2
<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1.1)</code></strong>
= -1
@@ -28,12 +46,12 @@ for more information.
exact arithmetic.
</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>Η εκθετική συνάρτηση. Αυτή είναι η συνάρτηση <strong
class="userinput"><code>e^x</code></strong> όπου <code class="varname">e</code> είναι η <a class="link"
href="ch11s04.html#gel-function-e">βάση του φυσικού λογαρίθμου</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Δημιουργεί αριθμό τιμής κινητής υποδιαστολής. Δηλαδή επιστρέφει την
αναπαράσταση κινητής υποδιαστολής του αριθμού <code class="varname">x</code>.</p></dd><dt><span
class="term"><a name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor
(x)</pre><p>Παραλλαγές: <code class="function">Floor</code></p><p>Παίρνει τον μεγαλύτερο ακέραιο που είναι
μικρότερος από ή ίσος με <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-ln"></a>ln</span></dt><dd><pre class="synopsis">ln (x)</pre><p>Ο φυσικός λογάριθμος, ο
λογάριθμος με βάση το <code class="varname">e</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Ο λογάριθμος του <code
class="varname">x</code> με βάση <code class="varname">b</code> (καλεί <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> αν είναι σε
κατάσταση modulo), αν η βάση δεν δίνεται, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> χρησιμοποιείται.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Ο λογάριθμος της
<code class="varname">x</code> με βάση 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Παραλλαγές: <code
class="function">lg</code></p><p>Ο λο�
�άριθμος του <code class="varname">x</code> με βάση 2.</p></dd><dt><span class="term"><a
name="gel-function-max"></a>max</span></dt><dd><pre class="synopsis">max (a,args...)</pre><p>Παραλλαγές:
<code class="function">Max</code><code class="function">Maximum</code></p><p>Επιστρέφει το μέγιστο των
ορισμάτων ή πίνακα.</p></dd><dt><span class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre
class="synopsis">min (a,args...)</pre><p>Παραλλαγές: <code class="function">Min</code><code
class="function">Minimum</code></p><p>Επιστρέφει το ελάχιστο των ορισμάτων ή πίνακα.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(size...)</pre><p>Δημιουργεί τυχαίο αριθμό κινητής υποδιαστολής στην περιοχή <code
class="literal">[0,1)</code>. Αν το μέγεθ�
�ς δίνεται, τότε ένας πίνακας (αν δύο αριθμοί ορίζονται) ή ένα διάνυσμα (αν ορίζεται ένας αριθμός) του
δοσμένου μεγέθους επιστρέφεται.</p></dd><dt><span class="term"><a
name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,size...)</pre><p>Δημιουργεί τυχαίο ακέραιο στην περιοχή <code class="literal">[0,max)</code>. Αν το
μέγεθος δίνεται, τότε ένας πίνακας (αν ορίζονται δύο αριθμοί) ή ένα διάνυσμα (αν ορίζεται ένας αριθμός) του
δοσμένου μεγέθους επιστρέφεται. Για παράδειγμα, </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>randint(4)</code></strong>
diff --git a/help/el/html/ch11s06.html b/help/el/html/ch11s06.html
index c48c58b..c4f293b 100644
--- a/help/el/html/ch11s06.html
+++ b/help/el/html/ch11s06.html
@@ -1,13 +1,21 @@
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title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της GEL"><link rel="prev" href="ch11s05.html"
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text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Τριγωνομετρία</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s05.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11.
Κατάλογος συναρτήσεων της GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Τριγωνομετρία</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Παραλλαγές: <code
class="function">arccos</code></p><p>Η συνάρτηση arccos (τόξο συνημιτόνου).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Παραλλαγές: <code
class="function">arccosh</code></p><p>Η συνάρτηση arccosh (τόξο υπερβολικού συνημιτόνου).</p></dd><dt><span
class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot
(x)</pre><p>Παραλλαγές: <code class="fu
nction">arccot</code></p><p>Η συνάρτηση arccot (τόξο συνεφαπτομένης).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Παραλλαγές: <code
class="function">arccoth</code></p><p>Η συνάρτηση arccoth (τόξο υπερβολικής
συνεφαπτομένης).</p></dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre
class="synopsis">acsc (x)</pre><p>Παραλλαγές: <code class="function">arccsc</code></p><p>Η συνάρτηση τόξου
συντέμνουσας.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Παραλλαγές: <code class="function">arccsch</code></p><p>Η συνάρτηση τόξου
υπερβολικής συντέμνουσας.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre
class="synopsis">asec (x)</pre><p>Παραλλαγές: <code class="function">arcsec</code></p><p>Η συνάρτηση τόξου
τέμνουσας.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Παραλλαγές: <code class="function">arcsech</code></p><p>Η συνάρτηση τόξου
υπερβολικής τέμνουσας.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin (x)</pre><p>Παραλλαγές: <code
class="function">arcsin</code></p><p>Η συνάρτηση arcsin (τόξο ημιτόνου).</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Παραλλαγές: <code
class="function">arcsinh</code></p><p>Η συνάρτηση arcsinh (τόξο υπερβολικού ημιτόνου).</p></dd><dt><span
class="term"><a name="gel-functio
n-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Παραλλαγές: <code
class="function">arctan</code></p><p>Υπολογίζει τη συνάρτηση arctan (τόξο εφαπτομένης).</p><p>Δείτε <a
class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Παραλλαγές: <code class="function">arctanh</code></p><p>Η συνάρτηση
arctanh (τόξο υπερβολικής εφαπτομένης).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Παραλλαγές:
<code class="function">arctan2</code></p><p>Calculates the arctan2 f
unction. If
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align="left"><a accesskey="p" href="ch11s05.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11.
Κατάλογος συναρτήσεων της GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Τριγωνομετρία</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Παραλλαγές: <code
class="function">arccos</code></p><p>Η συνάρτηση arccos (τόξο συνημιτόνου).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Παραλλαγές: <code
class="function">arccosh</code></p><p>Η συνάρτηση arccosh (τόξο υπερβολικού συνημιτόνου).</p></dd><dt><span
class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot
(x)</pre><p>Παραλλαγές: <code class="fu
nction">arccot</code></p><p>Η συνάρτηση arccot (τόξο συνεφαπτομένης).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Παραλλαγές: <code
class="function">arccoth</code></p><p>Η συνάρτηση arccoth (τόξο υπερβολικής
συνεφαπτομένης).</p></dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre
class="synopsis">acsc (x)</pre><p>Παραλλαγές: <code class="function">arccsc</code></p><p>Η συνάρτηση τόξου
συντέμνουσας.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Παραλλαγές: <code class="function">arccsch</code></p><p>Η συνάρτηση τόξου
υπερβολικής συντέμνουσας.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre
class="synopsis">asec (x)</pre><p>Παραλλαγές: <code class="function">arcsec</code></p><p>Η συνάρτηση τόξου
τέμνουσας.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Παραλλαγές: <code class="function">arcsech</code></p><p>Η συνάρτηση τόξου
υπερβολικής τέμνουσας.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin (x)</pre><p>Παραλλαγές: <code
class="function">arcsin</code></p><p>Η συνάρτηση arcsin (τόξο ημιτόνου).</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Παραλλαγές: <code
class="function">arcsinh</code></p><p>Η συνάρτηση arcsinh (τόξο υπερβολικού ημιτόνου).</p></dd><dt><span
class="term"><a name="gel-functio
n-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Παραλλαγές: <code
class="function">arctan</code></p><p>Υπολογίζει τη συνάρτηση arctan (τόξο εφαπτομένης).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Παραλλαγές: <code class="function">arctanh</code></p><p>Η συνάρτηση
arctanh (τόξο υπερβολικής εφαπτομένης).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Παραλλαγές:
<code class="function">arctan2</code></p><p>Calculates the arctan2 function. If
<strong class="userinput"><code>x>0</code></strong> then it returns
<strong class="userinput"><code>atan(y/x)</code></strong>. If <strong
class="userinput"><code>x<0</code></strong>
then it returns <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>.
When <strong class="userinput"><code>x=0</code></strong> it returns <strong
class="userinput"><code>sign(y) *
pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> returns 0
rather than failing.
- </p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a>
ή <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> για
περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-cos"></a>cos</span></dt><dd><pre class="synopsis">cos (x)</pre><p>Υπολογίζει τη συνάρτηση
του συνημιτόνου.</p><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Υπολογίζει τη συνάρτηση του συνημιτόνου.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Υπολογίζει την συνάρτηση υπερβολικού συνημιτόνου.</p><p>
See
@@ -15,7 +23,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>Η συνάρτηση συνεφαπτομένης.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Η συνάρτηση υπερβολικής συνεφαπτομένης.</p><p>
See
@@ -23,7 +31,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>Η συνάρτηση συντέμνουσας.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Η συνάρτηση υπερβολικής συντέμνουσας.</p><p>
See
@@ -31,7 +39,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Η συνάρτηση τέμνουσας.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Η συνάρτηση υπερβολικής τέμνουσας.</p><p>
See
@@ -39,7 +47,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Υπολογίζει τη συνάρτηση του ημιτόνου.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Υπολογίζει την συνάρτηση υπερβολικού ημιτόνου.</p><p>
See
@@ -47,7 +55,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Υπολογίζει τη συνάρτηση της εφαπτομένης.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Η συνάρτηση υπερβολικής εφαπτομένης.</p><p>
See
diff --git a/help/el/html/ch11s07.html b/help/el/html/ch11s07.html
index 5306b5e..68a1c69 100644
--- a/help/el/html/ch11s07.html
+++ b/help/el/html/ch11s07.html
@@ -3,36 +3,50 @@
<a class="ulink" href="https://en.wikipedia.org/wiki/Coprime_integers";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Επιστρέφει τον <code class="varname">n</code>στό αριθμό Bernoulli.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/BernoulliNumber.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Παραλλαγές: <code class="function">CRT</code></p><p>Εύρεση
του <code class="varname">x</code> που επιλύει το δοσμένο σύστημα με το διάνυσμα <code
class="varname">a</code> και modulo τα στοιχεία του <code class="varname">m</code>, χ
ρησιμοποιώντας το θεώρημα υπολοίπου του Κινέζου.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Επιστρέφει τον <code class="varname">n</code>στό αριθμό Bernoulli.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Παραλλαγές: <code class="function">CRT</code></p><p>Εύρεση
του <code class="varname">x</code> που επιλύει το δοσμένο σύστημα με το διάνυσμα <code
class="varname">a</code> και modulo τα στοιχεία του <code class="varname">m</code>, χρησιμοποιώντας το
θεώρημα υπολοίπου του Κινέζου.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Με δεδομένες δύο παραγοντοποιήσεις, δίνει την
παραγοντοποίηση του γινομένου.</p><p>Δείτε <a class="link"
href="ch11s07.html#gel-function-Factorize">παραγοντοποίηση</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Μετατρέπει ένα διάνυσμα τιμών που δείχνει δυνάμεις του b στον αριθμό a.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Μετατρέπει έναν αριθμό σε ένα διάνυσμα δυνάμεων για στοιχεία στ�
� βάση <code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>Βρίσκει τον διακριτό λογάριθμο της <code class="varname">n</code> με βάση <code
class="varname">b</code> στην F<sub>q</sub>, το πεπερασμένο πεδίο τάξης <code class="varname">q</code>, όπου
η <code class="varname">q</code> είναι ένας πρώτος, χρησιμοποιώντας τον αλγόριθμο
Silver-Pohlig-Hellman.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Ελέγχει τη διαιρετότητα (αν η <code class="varname">m</code> διαιρεί
την <code class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi
(n)</pre><p>Υπολογίζει τη συνάρτηση φι του Όιλερ για την <code class="varname">n</code>, δηλαδή τον αριθμό
των ακεραίων μεταξύ 1 και <code class="varname">n</code> που είναι σχετικά πρώτοι με την <code
class="varname">n</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Επιστρέφει <strong class="userinput"><code>n/d</code></strong>, αλλά μόνο αν η <code
class="varname">d</code> διαιρεί την <code class="varname">n</code>. Αν η <code class="varname">d</code> δεν
διαιρεί την <code class="varname">n</code>, τότε αυτή η συνάρτηση επιστρέφει σκουπίδια. Αυτή είναι πολύ πιο
γρήγορη για πολύ μεγάλους αριθμούς από την πράξη <strong class="userinput"><code>n/d</code></strong>, αλλά
φυσικά χρήσιμη μόνο αν ξέρετε ότι η διαίρεση είναι ακριβής.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize
(n)</pre><p>Επιστρέφει την παρ
αγοντοποίηση ενός αριθμού ως πίνακα. Η πρώτη γραμμή είναι οι πρώτοι στην παραγοντοποίηση (συμπεριλαμβάνοντας
το 1) και η δεύτερη γραμμή είναι οι δυνάμεις. Έτσι για παράδειγμα: </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>Factorize(11*11*13)</code></strong>
=
[1 11 13
- 1 2 1]</pre><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-Factors"></a>Factors</span></dt><dd><pre class="synopsis">Factors (n)</pre><p>Επιστρέφει
όλους τους παράγοντες της <code class="varname">n</code> σε ένα διάνυσμα. Αυτή περιλαμβάνει όλους τους μη
πρώτους παράγοντες επίσης. Περιλαμβάνει το 1 και τον ίδιο τον αριθμό. Έτσι για παράδειγμα για να εμφανίσετε
όλους τους τέλειους αριθμούς (αυτούς που είναι αθροίσματα των παραγόντων τους) μέχρι τον αριθμό 1000 μπορείτε
να κάνετε (αυτό φυσικά είναι πολύ ανεπαρκές) </p><
pre class="programlisting">for n=1 to 1000 do (
+ 1 2 1]</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>Επιστρέφει όλους τους παράγοντες της <code class="varname">n</code> σε
ένα διάνυσμα. Αυτή περιλαμβάνει όλους τους μη πρώτους παράγοντες επίσης. Περιλαμβάνει το 1 και τον ίδιο τον
αριθμό. Έτσι για παράδειγμα για να εμφανίσετε όλους τους τέλειους αριθμούς (αυτούς που είναι αθροίσματα των
παραγόντων τους) μέχρι τον αριθμό 1000 μπορείτε να κάνετε (αυτό φυσικά είναι πολύ ανεπαρκές) </p><pre
class="programlisting">for n=1 to 1000 do (
if MatrixSum (Factors(n)) == 2*n then
print(n)
)
-</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,tries)</pre><p>Δοκιμάζει την παραγοντοποίηση Φερμά της <code
class="varname">n</code> στο <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, επιστρέφει τις
<code class="varname">t</code> και <code class="varname">s</code> ως ένα διάνυσμα αν είναι δυνατό, αλλιώς
<code class="constant">null</code>. Η <code class="varname">tries</code> καθορίζει τον αριθμό των προσπαθειών
πριν να σταματήσει.</p><p>Αυτή είναι μια αρκετά καλή παραγοντοποίηση, αν ο αριθμός σας είναι το γινόμενο δύο
παραγόντων που είναι πολύ κοντά μεταξύ τους.</p><p>Δείτε <a class="ulink" href="http://en.wi
kipedia.org/wiki/Fermat_factorization" target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Βρίσκει το πρώτο βασικό στοιχείο στην F<sub>q</sub>, την
πεπερασμένη ομάδα της τάξης <code class="varname">q</code>. Φυσικά η <code class="varname">q</code> πρέπει να
είναι πρώτος.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Βρίσκει ένα τυχαίο βασικό στοιχείο στην
F<sub>q</sub>, την πεπερασμένη ομάδα της τάξης <code class="varname">q</code> (το q πρέπει να είναι πρώτ
ος).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Υπολογίζει τη διακριτή λογαριθμική βάση <code class="varname">b</code> του n στο
F<sub>q</sub>, την πεπερασμένη ομάδα της τάξης <code class="varname">q</code> (<code class="varname">q</code>
είναι ένας πρώτος), χρησιμοποιώντας τη βάση του συντελεστή <code class="varname">S</code>. Η <code
class="varname">S</code> πρέπει να είναι μια στήλη πρώτων πιθανόν με μια δεύτερη στήλη προϋπολογισμένη από
την <a class="link" href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalcu
lusPrecalculation</span></dt><dd><pre class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Εκτελεί
το βήμα προϋπολογισμού του <a class="link" href="ch11s07.html#gel-function-IndexCalculus"><code
class="function">IndexCalculus</code></a> για λογαρίθμους με βάση <code class="varname">b</code> στο
F<sub>q</sub>, την πεπερασμένη ομάδα της τάξης <code class="varname">q</code> (<code class="varname">q</code>
είναι ένας πρώτος), για τη βάση συντελεστή <code class="varname">S</code> (όπου <code
class="varname">S</code> είναι ένα διάνυσμα στήλης πρώτων). Οι λογάριθμοι θα προϋπολογιστούν και θα
επιστραφούν στη δεύτερη στήλη.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven (n)</pre><p>Ελέγχει αν ο �
�κέραιος είναι άρτιος.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Ελέγχει αν ο θετικός ακέραιος <code
class="varname">p</code> είναι ένας εκθέτης πρώτου Μερσέν. Δηλαδή, αν 2<sup>p</sup>-1 είναι ένας πρώτος. Το
κάνει αναζητώντας τον σε έναν πίνακα γνωστών τιμών που είναι σχετικά σύντομος. Δείτε επίσης <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> και <a class="link"
href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
+</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,tries)</pre><p>Δοκιμάζει την παραγοντοποίηση Φερμά της <code
class="varname">n</code> στο <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, επιστρέφει τις
<code class="varname">t</code> και <code class="varname">s</code> ως ένα διάνυσμα αν είναι δυνατό, αλλιώς
<code class="constant">null</code>. Η <code class="varname">tries</code> καθορίζει τον αριθμό των προσπαθειών
πριν να σταματήσει.</p><p>Αυτή είναι μια αρκετά καλή παραγοντοποίηση, αν ο αριθμός σας είναι το γινόμενο δύο
παραγόντων που είναι πολύ κοντά μεταξύ τους.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Βρίσκει το πρώτο βασικό στοιχείο στην F<sub>q</sub>, την
πεπερασμένη ομάδα της τάξης <code class="varname">q</code>. Φυσικά η <code class="varname">q</code> πρέπει να
είναι πρώτος.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Βρίσκει ένα τυχαίο βασικό στοιχείο στην
F<sub>q</sub>, την πεπερασμένη ομάδα της τάξης <code class="varname">q</code> (το q πρέπει να είναι
πρώτος).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class=
"synopsis">IndexCalculus (n,b,q,S)</pre><p>Υπολογίζει τη διακριτή λογαριθμική βάση <code
class="varname">b</code> του n στο F<sub>q</sub>, την πεπερασμένη ομάδα της τάξης <code
class="varname">q</code> (<code class="varname">q</code> είναι ένας πρώτος), χρησιμοποιώντας τη βάση του
συντελεστή <code class="varname">S</code>. Η <code class="varname">S</code> πρέπει να είναι μια στήλη πρώτων
πιθανόν με μια δεύτερη στήλη προϋπολογισμένη από την <a class="link"
href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Εκτελεί το
βήμα προϋπολογισμού του <a class="link" href="ch11s07.html#gel-function-IndexCalculus"><code
class="function">IndexCalculus</code></a> για λογαρίθμους με βάση <code class="varname">b</code> στο
F<sub>q</sub>, την πεπερασμένη ομάδα της τάξης <code class="varname">q</code> (<code class="varname">q</code>
είναι ένας πρώτος), για τη βάση συντελεστή <code class="varname">S</code> (όπου <code
class="varname">S</code> είναι ένα διάνυσμα στήλης πρώτων). Οι λογάριθμοι θα προϋπολογιστούν και θα
επιστραφούν στη δεύτερη στήλη.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven (n)</pre><p>Ελέγχει αν ο
ακέραιος είναι άρτιος.</p></dd><dt><span class="term"><a name="gel-function-IsMersennePrimeExponent">
</a>IsMersennePrimeExponent</span></dt><dd><pre class="synopsis">IsMersennePrimeExponent (p)</pre><p>Ελέγχει
αν ο θετικός ακέραιος <code class="varname">p</code> είναι ένας εκθέτης πρώτου Μερσέν. Δηλαδή, αν
2<sup>p</sup>-1 είναι ένας πρώτος. Το κάνει αναζητώντας τον σε έναν πίνακα γνωστών τιμών που είναι σχετικά
σύντομος. Δείτε επίσης <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> και <a class="link"
href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Ελέγχει αν ένας ρητός αριθμός <code class="varname">m</code> είναι μια τέλεια <code
class="varname">n</code>στή δύναμη. Δείτε επίσης <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> και <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Έλεγχος αν ο
ακέραιος είναι περιττός.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare
"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare (n)</pre><p>Ελέγχει αν ένας
ακέραιος είναι τέλειο τετράγωνο ενός ακεραίου. Ο αριθμός πρέπει να είναι πραγματικός ακέραιος. Οι αρνητικοί
ακέραιοι φυσικά δεν είναι ποτέ τέλεια τετράγωνα πραγματικών ακεραίων.</p></dd><dt><span class="term"><a
name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre class="synopsis">IsPrime (n)</pre><p>Ελέγχει τους
πρώτους αριθμούς ακεραίων, για αριθμούς μικρότερους από 2.5e10 η απάντηση είναι προσδιοριστική (αν η υπόθεση
Ρίμαν είναι αληθής). Για αριθμούς μεγαλύτερους, η πιθανότητα ψευδών θετικών εξαρτάται από την <a class="link"
href="
ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code class="function">IsPrimeMillerRabinReps</code></a>.
Δηλαδή, η πιθανότητα ψευδούς θετικού είναι 1/4 στη δύναμη <code
class="function">IsPrimeMillerRabinReps</code>. Η προεπιλεγμένη τιμή του 22 δίνει μια πιθανότητα περίπου
5.7e-14.</p><p>Αν η <code class="constant">ψευδής</code> επιστρέφεται, μπορείτε να είστε σίγουροι ότι ο
αριθμός είναι σύνθετος. Αν θέλετε να είσαστε ολότελα βέβαιοι ότι έχετε έναν πρώτο μπορείτε να χρησιμοποιήσετε
την <a class="link" href="ch11s07.html#gel-function-MillerRabinTestSure"><code
class="function">MillerRabinTestSure</code></a>, αλλά μπορεί να πάρει πολύ περισσότερο.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Ελέγχει αν ένας ρητός αριθμός <code class="varname">m</code> είναι μια τέλεια <code
class="varname">n</code>στή δύναμη. Δείτε επίσης <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> και <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Έλεγχος αν ο
ακέραιος είναι περιττός.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare
"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare (n)</pre><p>
+ Check an integer for being a perfect square of an integer. The number must
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
+ </p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>Ελέγχει τους πρώτους αριθμούς ακεραίων, για αριθμούς μικρότερους από
2.5e10 η απάντηση είναι προσδιοριστική (αν η υπόθεση Ρίμαν είναι αληθής). Για αριθμούς μεγαλύτερους, η
πιθανότητα ψευδών θετικών εξαρτάται από την <a class="link"
href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. Δηλαδή, η πιθανότητα ψευδούς θετικού είναι 1/4 στη δύναμη
<code class="function">IsPrimeMillerRabinReps</code>. Η προεπιλεγμένη τιμή του 22 δίνει μια πιθανότητα
περίπου 5.7e-14.</p><p>Αν η <code class="constant">ψευδής</code> επι�
�τρέφεται, μπορείτε να είστε σίγουροι ότι ο αριθμός είναι σύνθετος. Αν θέλετε να είσαστε ολότελα βέβαιοι ότι
έχετε έναν πρώτο μπορείτε να χρησιμοποιήσετε την <a class="link"
href="ch11s07.html#gel-function-MillerRabinTestSure"><code class="function">MillerRabinTestSure</code></a>,
αλλά μπορεί να πάρει πολύ περισσότερο.</p><p>
See
<a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a>
for more information.
@@ -42,24 +56,24 @@
<a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Ελέγχει αν 2<sup>p</sup>-1 είναι ένας πρώτος Μερσέν χρησιμοποιώντας τη δοκιμή Lucas-Lehmer. Δείτε
επίσης <a class="link" href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> και
<a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Επιστρέφει τον <code class="varname">n</code>στο αριθμό Lucas.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Επιστρέφει όλους τους μέγιστους πρώτους παράγοντες
δύναμης ενός αριθμού.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>Ένα διάνυσμα με γνωστούς πρώτους εκθέτες Μερσέν, δηλαδή ένας
κατάλογος θετικών ακεραίων <code class="varname">p</code> έτσι ώστε το 2<sup>p</sup>-1 να είναι πρώτος. Δείτε
επίσης <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>
και <a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre class="synopsis">MillerRabinTest
(n,reps)</pre><p>Χρησιμοποιεί τη δοκιμή πρώτων αριθμών Miller-Rabin στο <code class="varname">n</code>, <code
class="varname">reps</code> είναι ο αριθμός των φορών. Η πιθανότητα ενός ψευδούς θετικού είναι <strong
class="userinput"><code>(1/4)^reps</code></strong>. Είναι προφανώς συνήθως καλύτερο να χρησιμοποιήσετε <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a> αφού είναι
γρηγορότερο και καλύτερο σε μικρότερους ακέραιους.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
@@ -68,7 +82,7 @@
result is deterministic.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Επιστρέφει τον αντίστροφο του n mod m.</p><p>Δείτε <a class="ulink"
href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu
(n)</pre><p>Επιστρέφει την συνάρτηση mu του Moebius υπολογισμένη στο <code class="varname">n</code>. Δηλαδή,
επιστρέφει 0 αν το <code class="varname">n</code> δεν είναι γινόμενο διακριτών πρώτων και <strong
class="userinput"><code>(-1)^k</code></strong> αν είναι γινόμενο των <code class="varname">k</code> διακριτών
πρώτων.</p><p>
diff --git a/help/el/html/ch11s08.html b/help/el/html/ch11s08.html
index 2da51ad..4fb9b3a 100644
--- a/help/el/html/ch11s08.html
+++ b/help/el/html/ch11s08.html
@@ -1,4 +1,12 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Διαχείριση
πινάκων</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html" title="Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL"><link rel="prev" href="ch11s07.html" title="Θεωρία αριθμών"><link rel="next"
href="ch11s09.html" title="Γραμμική Άλγεβρα"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Διαχείριση πινάκων</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s07.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL</th><td width="20%" alig
n="right"> <a accesskey="n" href="ch11s09.html">Επόμενο</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Διαχείριση πινάκων</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Εφαρμόζει μια συνάρτηση σε όλες τις καταχωρίσεις ενός πίνακα και επιστρέφει έναν πίνακα των
αποτελεσμάτων.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Εφαρμόζει μια συνάρτηση σε όλες τις καταχωρίσεις των 2
πινάκων (ή 1 τιμή κα
ι 1 πίνακα) και επιστρέφει έναν πίνακα των αποτελεσμάτων.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf
(M)</pre><p>Παίρνει τις στήλες ενός πίνακα ως οριζόντιο διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre
class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Αφαιρεί στήλες και γραμμές από έναν
πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Υπολογίζει τον kστό σύνθετο πίνακα του Α.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>Υπολο�
�ίζει τον αριθμό των μηδενικών στηλών σε έναν πίνακα. Για παράδειγμα, αφού ο πίνακάς σας μειώσει έναν
πίνακα, μπορείτε να τον χρησιμοποιήσετε για να βρείτε τη μηδενικότητα. Δείτε <a class="link"
href="ch11s09.html#gel-function-cref"><code class="function">cref</code></a> και <a class="link"
href="ch11s09.html#gel-function-Nullity"><code class="function">Nullity</code></a>.</p></dd><dt><span
class="term"><a name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre
class="synopsis">DeleteColumn (M,col)</pre><p>Διαγράφει μια στήλη ενός πίνακα.</p></dd><dt><span
class="term"><a name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Διαγράφει μια γραμμή ενός πίνακα.</p></dd><dt><span class="term"><a name="gel-functio
n-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf (M)</pre><p>Δίνει τις διαγώνιες
καταχωρίσεις ενός πίνακα ως διάνυσμα στήλης.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices"; target="_top">Wikipedia</a> για
περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Διαχείριση
πινάκων</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html" title="Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL"><link rel="prev" href="ch11s07.html" title="Θεωρία αριθμών"><link rel="next"
href="ch11s09.html" title="Γραμμική Άλγεβρα"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Διαχείριση πινάκων</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s07.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL</th><td width="20%" alig
n="right"> <a accesskey="n" href="ch11s09.html">Επόμενο</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Διαχείριση πινάκων</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Εφαρμόζει μια συνάρτηση σε όλες τις καταχωρίσεις ενός πίνακα και επιστρέφει έναν πίνακα των
αποτελεσμάτων.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Εφαρμόζει μια συνάρτηση σε όλες τις καταχωρίσεις των 2
πινάκων (ή 1 τιμή κα
ι 1 πίνακα) και επιστρέφει έναν πίνακα των αποτελεσμάτων.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf
(M)</pre><p>Παίρνει τις στήλες ενός πίνακα ως οριζόντιο διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre
class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Αφαιρεί στήλες και γραμμές από έναν
πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Υπολογίζει τον kστό σύνθετο πίνακα του Α.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
+ the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
+ and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Διαγράφει μια στήλη ενός πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Διαγράφει μια γραμμή ενός πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Δίνει τις διαγώνιες καταχωρίσεις ενός πίνακα ως διάνυσμα στήλης.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Dot_product"; target="_top">Wikipedia</a> or
@@ -13,11 +21,17 @@
<a class="ulink" href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vec,msize)</pre><p>Επιστρέφει το συμπλήρωμα δείκτη ενός διανύσματος δεικτών. Ο πρώτος δείκτης είναι ένα. Για
παράδειγμα για διάνυσμα <strong class="userinput"><code>[2,3]</code></strong> και μέγεθος <strong
class="userinput"><code>5</code></strong>, επιστρέφει <strong
class="userinput"><code>[1,4,5]</code></strong>. Αν <code class="varname">msize</code> είναι 0, επιστρέφει
πάντα <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Αν
είναι ένας διαγώνιος πίνακας.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Ελέγχει αν ένας πίνακας είναι ο ταυτοτικός πίνακας. Επιστρέφει αυτόματα <code
class="constant">ψευδές</code> αν ο πίνακας δεν είναι τετραγωνικός. Δουλεύει επίσης με αριθμούς και σε αυτήν
την περίπτωση είναι ισοδύναμος με <strong class="userinput"><code>x==1</code></strong>. Όταν <code
class="varname">x</code> είναι <code class="constant">null</code> (μπορούμε να τον θεωρήσουμε ως έναν πίνακα
0 επί 0), δεν δημιουργείται κανένα σφάλμα και επιστρέφεται <code
class="constant">ψευδές</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</sp
an></dt><dd><pre class="synopsis">IsLowerTriangular (M)</pre><p>Αν είναι ένας κάτω τριγωνικός πίνακας.
Δηλαδή, αν είναι όλες οι καταχωρίσεις πάνω από τη διαγώνιο είναι μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Ελέγχει αν ο πίνακας είναι μη αρνητικός, δηλαδή, αν κάθε
στοιχείο είναι μη αρνητικός. Μην μπερδεύετε θετικά πίνακες με θετικούς ημιορισμένους πίνακες.</p><p>Δείτε <a
class="ulink" href="http://en.wikipedia.org/wiki/
Positive_matrix" target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Ελέγχει αν ένας πίνακας είναι θετικός, δηλαδή, αν κάθε στοιχείο
είναι θετικό (και συνεπώς πραγματικό). Ειδικά, κανένα στοιχείο δεν είναι 0. Μην μπερδεύετε θετικούς πίνακες
με θετικά ορισμένους πίνακες.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Ελέγχει αν ένας πίν�
�κας είναι ένας πίνακας ρητών αριθμών (μη μιγαδικός).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Ελέγχει αν ένας πίνακας είναι ένας πίνακας πραγματικών αριθμών (μη μιγαδικός).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Ελέγχει αν ένας πίνακας είναι τετράγωνος, δηλαδή, αν το πλάτος
του είναι ίσο με το ύψος του.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Είναι ένας άνω τριγωνικός πίνακας; Δηλαδή, ένας πίνακας είναι
άνω τρ
ιγωνικός αν όλες οι καταχωρίσεις κάτω από τη διαγώνιο είναι μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Ελέγχει αν ο πίνακας είναι ένας πίνακας μόνο αριθμών. Πολλές εσωτερικές συναρτήσεις κάνουν αυτόν
τον έλεγχο. Οι τιμές μπορεί να είναι οποιοιδήποτε αριθμοί συμπεριλαμβανομένων μιγαδικών
αριθμών.</p></dd><dt><span class="term"><a name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre
class="synopsis">IsVector (v)</pre><p>Αν είναι το όρισμα οριζόντιο ή κάθετο διάνυσμα. Η Genius δεν ξεχωρίζει
μεταξύ πίνακα και διανύσματος και ένα διάνυσμα είναι απλά �
�νας πίνακας 1 επί <code class="varname">n</code> ή <code class="varname">n</code> επί 1.</p></dd><dt><span
class="term"><a name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero
(x)</pre><p>Ελέγχει αν ένας πίνακας αποτελείται όλος από μηδενικά. Δουλεύει επίσης και σε αριθμούς, οπότε
είναι ισοδύναμος με <strong class="userinput"><code>x==0</code></strong>. Όταν η <code
class="varname">x</code> είναι <code class="constant">null</code> (μπορούμε να σκεφτούμε ως έναν πίνακα 0 επί
0), δεν δημιουργείται κανένα σφάλμα και επιστρέφεται η <code class="constant">true</code> επειδή η συνθήκη
είναι κενή.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriang
ular (M)</pre><p>Επιστρέφει ένα αντίγραφο του πίνακα <code class="varname">M</code> με όλες τις καταχωρίσεις
πάνω από τη διαγώνιο ορισμένες σε μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Παραλλαγές: <code class="function">diag</code></p><p>Δημιουργεί έναν διαγώνιο πίνακα από
ένα διάνυσμα. Εναλλακτικά μπορείτε να περάσετε στις τιμές για να βάλετε τη διαγώνιο ως ορίσματα. Έτσι <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> είναι το ίδιο με <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Ελέγχει αν ένας πίνακας είναι ο ταυτοτικός πίνακας. Επιστρέφει αυτόματα <code
class="constant">ψευδές</code> αν ο πίνακας δεν είναι τετραγωνικός. Δουλεύει επίσης με αριθμούς και σε αυτήν
την περίπτωση είναι ισοδύναμος με <strong class="userinput"><code>x==1</code></strong>. Όταν <code
class="varname">x</code> είναι <code class="constant">null</code> (μπορούμε να τον θεωρήσουμε ως έναν πίνακα
0 επί 0), δεν δημιουργείται κανένα σφάλμα και επιστρέφεται <code
class="constant">ψευδές</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</sp
an></dt><dd><pre class="synopsis">IsLowerTriangular (M)</pre><p>Αν είναι ένας κάτω τριγωνικός πίνακας.
Δηλαδή, αν είναι όλες οι καταχωρίσεις πάνω από τη διαγώνιο είναι μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Ελέγχει αν ο πίνακας είναι μη αρνητικός, δηλαδή, αν κάθε
στοιχείο είναι μη αρνητικός. Μην μπερδεύετε θετικά πίνακες με θετικούς ημιορισμένους πίνακες.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Ελέγχει αν ένας πίνακας είναι θετικός, δηλαδή, αν κάθε στοιχείο
είναι θετικό (και συνεπώς πραγματικό). Ειδικά, κανένα στοιχείο δεν είναι 0. Μην μπερδεύετε θετικούς πίνακες
με θετικά ορισμένους πίνακες.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Ελέγχει αν ένας πίνακας είναι ένας πίνακας ρητών αριθμών (μη
μιγαδικός).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Ελέγχει αν ένας πίνακας είναι ένας πίνακας πραγματικών αριθμών (μη μιγαδικός).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Ελέγχει αν ένας πίνακας είναι τετράγωνος, δηλαδή, αν το πλάτος
του είναι ίσο με το ύψος του.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTria
ngular</span></dt><dd><pre class="synopsis">IsUpperTriangular (M)</pre><p>Είναι ένας άνω τριγωνικός πίνακας;
Δηλαδή, ένας πίνακας είναι άνω τριγωνικός αν όλες οι καταχωρίσεις κάτω από τη διαγώνιο είναι
μηδέν.</p></dd><dt><span class="term"><a name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre
class="synopsis">IsValueOnly (M)</pre><p>Ελέγχει αν ο πίνακας είναι ένας πίνακας μόνο αριθμών. Πολλές
εσωτερικές συναρτήσεις κάνουν αυτόν τον έλεγχο. Οι τιμές μπορεί να είναι οποιοιδήποτε αριθμοί
συμπεριλαμβανομένων μιγαδικών αριθμών.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Αν είναι
το όρισμα
οριζόντιο ή κάθετο διάνυσμα. Η Genius δεν ξεχωρίζει μεταξύ πίνακα και διανύσματος και ένα διάνυσμα είναι
απλά ένας πίνακας 1 επί <code class="varname">n</code> ή <code class="varname">n</code> επί
1.</p></dd><dt><span class="term"><a name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre
class="synopsis">IsZero (x)</pre><p>Ελέγχει αν ένας πίνακας αποτελείται όλος από μηδενικά. Δουλεύει επίσης
και σε αριθμούς, οπότε είναι ισοδύναμος με <strong class="userinput"><code>x==0</code></strong>. Όταν η <code
class="varname">x</code> είναι <code class="constant">null</code> (μπορούμε να σκεφτούμε ως έναν πίνακα 0 επί
0), δεν δημιουργείται κανένα σφάλμα και επιστρέφεται η <code class="constant">true
</code> επειδή η συνθήκη είναι κενή.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Επιστρέφει ένα αντίγραφο του πίνακα <code class="varname">M</code> με όλες τις καταχωρίσεις πάνω
από τη διαγώνιο ορισμένες σε μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Παραλλαγές: <code class="function">diag</code></p><p>Δημιουργεί έναν διαγώνιο πίνακα από
ένα διάνυσμα. Εναλλακτικά μπορείτε να περάσετε στις τιμές για να βάλετε τη διαγώνιο ως ορίσματα. Έτσι <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> είναι το �
�διο με <strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Δημιουργεί ένα διάνυσμα στήλης από έναν πίνακα βάζοντας στήλες τις μεν πάνω από τις άλλες.
Επιστρέφει <code class="constant">null</code> όταν δίνεται <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>Υπολογίζει το γινόμενο όλων των στοιχείων σε ένα πίνακα ή διάνυσμα. Δηλαδή, πολλαπλασιάζουμε όλα
τα στοιχεία και επιστρέφει έναν αριθμό που είναι το γινόμενο όλων των στοιχείων.</p></dd><dt><span
class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</sp
an></dt><dd><pre class="synopsis">MatrixSum (A)</pre><p>Υπολογίζει το άθροισμα όλων των στοιχείων σε ένα
πίνακα ή διάνυσμα. Δηλαδή, προσθέτουμε όλα τα στοιχεία και επιστρέφει έναν αριθμό που είναι το άθροισμα όλων
των στοιχείων.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Υπολογίζει το άθροισμα των τετραγώνων όλων των στοιχείων σε
έναν πίνακα ή διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Επιστρέφει ένα διάνυσμα γραμμής των δεικτών των μη μηδενικών στηλών στον πίνακα <code
class="varname">M</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Επιστρέφει ένα διάνυσμα γραμμής των δεικτών των μη μηδενικών στοιχείων του διανύσματος <code
class="varname">v</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Δίνει το εξωτερικό γινόμενο δύο διανυσμάτων. Δηλαδή, ας υποθέσουμε ότι <code
class="varname">u</code> και <code class="varname">v</code> είναι κάθετα διανύσματα, τότε το εξωτερικό
γινόμενο είναι <strong class="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span class="term"><a
name="ge
l-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Αντίστροφα στοιχεία σε ένα διάνυσμα. Επιστρέφει <code class="constant">null</code> αν δίνεται
<code class="constant">null</code></p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Υπολογίζει το
άθροισμα κάθε γραμμής σε έναν πίνακα και επιστρέφει ένα κάθετο διάνυσμα με το αποτέλεσμα.</p></dd><dt><span
class="term"><a name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre
class="synopsis">RowSumSquares (m)</pre><p>Υπολογίζει το άθροισμα των τετραγώνων κάθε γραμμής σε έναν πίνακα
και επιστρέφει ένα κάθετο διάνυσμα με τα αποτελέσματα.</p></dd><dt>
<span class="term"><a name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre class="synopsis">RowsOf
(M)</pre><p>Δίνει τις γραμμές ενός πίνακα ως κάθετο διάνυσμα. Κάθε στοιχείο του διανύσματος είναι ένα
οριζόντιο διάνυσμα που είναι η αντίστοιχη γραμμή του <code class="varname">M</code>. Αυτή η συνάρτηση είναι
χρήσιμη, αν θέλετε να κάνετε βρόχο στις γραμμές ενός πίνακα. Για παράδειγμα, ως <strong
class="userinput"><code>for r in RowsOf(M) do
something(r)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-SetMatrixSize"></a>SetMatrixSize</span></dt><dd><pre class="synopsis">SetMatrixSize
(M,rows,columns)</pre><p>Δημιουργεί νέο πίνακα δεδομένου μεγέθους από τον παλιό. Δηλαδή, θα επιστραφεί ένας
νέος πίνακας στον οποίον ο παλιός αντιγράφηκε. Οι καταχωρίσεις που δεν ταιριάζουν περικόπτονται και ο
πρόσθετος χώρος συμπληρώνεται με μηδενικά. Αν <code class="varname">rows</code> ή <code
class="varname">columns</code> είναι μηδέν, τότε επιστρέφεται <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-ShuffleVector"></a>ShuffleVector</span></dt><dd><pre class="synopsis">ShuffleVector
(v)</pre><p>Shuffle elements in a vector. Return <code class="const
ant">null</code> if given <code class="constant">null</code>.</p><p>Version 1.0.13
onwards.</p></dd><dt><span class="term"><a name="gel-function-SortVector"></a>SortVector</span></dt><dd><pre
class="synopsis">SortVector (v)</pre><p>Ταξινόμηση στοιχείων διανύσματος με αύξουσα
διάταξη.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroColumns"></a>StripZeroColumns</span></dt><dd><pre
class="synopsis">StripZeroColumns (M)</pre><p>Αφαιρεί όλες τις ολότελα μηδενικές στήλες του <code
class="varname">M</code>.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroRows"></a>StripZeroRows</span></dt><dd><pre class="synopsis">StripZeroRows
(M)</pre><p>Αφαιρεί όλες τις ολότελα μηδενικές γραμμές του <code class="varname">M</code>.</p></dd><dt><span
class="term"><a name="gel-function-Submatrix"></a>Submatrix</span></dt><dd><pre class="synopsis">
Submatrix (m,r,c)</pre><p>Επιστρέφει στήλες και γραμμές από έναν πίνακα. Αυτό είναι ακριβώς ισοδύναμο με το
<strong class="userinput"><code>m@(r,c)</code></strong>. Τα <code class="varname">r</code> και <code
class="varname">c</code> πρέπει να είναι διανύσματα γραμμών και στηλών (ή μεμονωμένοι αριθμοί αν χρειάζεται
μόνο μια γραμμή ή στήλη).</p></dd><dt><span class="term"><a
name="gel-function-SwapRows"></a>SwapRows</span></dt><dd><pre class="synopsis">SwapRows
(m,row1,row2)</pre><p>Εναλλάσσει δύο γραμμές σε έναν πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-UpperTriangular"></a>UpperTriangular</span></dt><dd><pre class="synopsis">UpperTriangular
(M)</pre><p>Επιστρέφει ένα αντίγραφο του πίνακα <code class="varname">M</code> με όλες τ
ις καταχωρίσεις κάτω από τη διαγώνιο ορισμένες σε μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-columns"></a>columns</span></dt><dd><pre class="synopsis">columns (M)</pre><p>Δίνει τον
αριθμό των στηλών ενός πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-elements"></a>elements</span></dt><dd><pre class="synopsis">elements (M)</pre><p>Δίνει τον
συνολικό αριθμό των στοιχείων ενός πίνακα. Αυτός είναι ο αριθμός των στηλών επί τον αριθμό των
γραμμών.</p></dd><dt><span class="term"><a name="gel-function-ones"></a>ones</span></dt><dd><pre
class="synopsis">ones (rows,columns...)</pre><p>Δημιουργεί έναν πίνακα από όλους (ή ένα διάνυσμα γραμμής αν
δίνεται μόνο ένα όρισμα). Επιστρέφει <code class="constant">null
</code> αν οποιαδήποτε σειρά ή στήλη είναι μηδέν.</p></dd><dt><span class="term"><a
name="gel-function-rows"></a>rows</span></dt><dd><pre class="synopsis">rows (M)</pre><p>Δίνει τον αριθμό των
γραμμών ενός πίνακα.</p></dd><dt><span class="term"><a
name="gel-function-zeros"></a>zeros</span></dt><dd><pre class="synopsis">zeros
(rows,columns...)</pre><p>Δημιουργεί έναν πίνακα όλων των μηδενικών (ή ένα διάνυσμα γραμμής αν δίνεται μόνο
ένα όρισμα). Επιστρέφει <code class="constant">null</code> αν οποιαδήποτε σειρά ή στήλη είναι
μηδέν.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s07.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Πάνω
</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s09.html">Επόμενο</a></td></tr><tr><td
width="40%" align="left" valign="top">Θεωρία αριθμών </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Αρχή</a></td><td width="40%" align="right" valign="top"> Γραμμική
Άλγεβρα</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s09.html b/help/el/html/ch11s09.html
index 5a17347..b37578c 100644
--- a/help/el/html/ch11s09.html
+++ b/help/el/html/ch11s09.html
@@ -28,12 +28,12 @@
<a class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Παραλλαγές: <code class="function">eig</code></p><p>Δίνει τις ιδιοτιμές ενός τετραγωνικού πίνακα.
Προς το παρόν δουλεύει μόνο για πίνακες μεγέθους μέχρι 4 επί 4, ή για τριγωνικούς πίνακες (για τους οποίους
οι ιδιοτιμές είναι στη διαγώνιο).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Δίνει τα ιδιοδιανύσματα ενός τετραγωνικού πίνακα. Προαιρετικά
παίρνετε επίσης τις ιδιοτιμές και τις αλγεβρικές πολλαπλότητες. Προς το παρόν δουλεύει μόνο για πίνακες
μεγέθους μέχρι 2 επί 2.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Εφαρμόζει τη διεργασία Gram-Schmidt (στις στήλες) ως προς το εσωτερικό γινόμενο που δίνεται
από το <code class="varname">B</code>. Αν το <code class="varname">B</code> δεν δίνεται, τότε χρησιμοποιείται
το τυπικό ερμιτιανό γινόμενο. Το <code class="varname">B</code> μπορεί να είναι ή γραμμικο-ημιγραμμική
συνάρτηση δύο ορισμάτων ή μπορεί να είναι ένας πίνακας που δίνει μια γραμμικο-ημιγραμμική μορφή. Τα
διανύσματα θα γίνονται ορθοκανονικά ως προς το <code class="varname">B</code>.</p><p>
@@ -86,7 +86,7 @@
<a class="ulink" href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre
class="synopsis">Kernel (T)</pre><p>Δίνει τον πυρήνα (διάστημα κενού) ενός γραμμικού
μετασχηματισμού.</p><p>(Δείτε <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Παραλλαγές: <code
class="function">TensorProduct</code></p><p>Υπολογίζει το γινόμενο Κρόνεκερ (γινόμενο τανυστή σε τυπική βάση)
δύο πινάκων.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
@@ -104,7 +104,7 @@
<code class="varname">U</code>.
</p><p>Αυτή είναι η ανάλυση LU ενός πίνακα γνωστό και ως Crout και/ή αναγωγή Σολεσκί. (ISBN
0-201-11577-8 pp.99-103) Ο άνω τριγωνικός πίνακας χαρακτηρίζει μια διαγώνιο τιμών 1 (ένα). Αυτή δεν είναι η
μέθοδος του Doolittle που χαρακτηρίζει τη διαγώνιο του 1 στον κάτω πίνακα.</p><p>Δεν έχουν όλοι οι πίνακες
αναλύσεις LU, για παράδειγμα το <strong class="userinput"><code>[0,1;1,0]</code></strong> δεν έχει και αυτή η
συνάρτηση επιστρέφει <code class="constant">false</code> σε αυτήν την περίπτωση και ορίζει <code
class="varname">L</code> και<code class="varname">U</code> σε <code class="constant">null</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Δίνει τον ελάσσονα <code class="varname">i</code>-<code
class="varname">j</code> ενός πίνακα.</p><p>
@@ -119,7 +119,7 @@
</p></dd><dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre><p>Δίνει του ορθογωνίου συμπληρώματος του χώρου
στήλης.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Επιστρέφει τις οδηγούσες στήλες ενός πίνακα, δηλαδή τις στήλες που έχουν ένα αρχικό 1 σε ανηγμένη
μορφή κατά γραμμές. Επίσης επιστρέφει τη γραμμή που αυτό συμβαίνει.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Προβολή του διανύσματος <code class="varname">v</code> στον υπόχωρο <code class="va
rname">W</code> ως προς το δοσμένο εσωτερικό γινόμενο από το <code class="varname">B</code>. Αν το <code
class="varname">B</code> δεν δίνεται, τότε το τυπικό ερμιτιανό γινόμενο χρησιμοποιείται. Το <code
class="varname">B</code> μπορεί ή να είναι γραμμικο-ημιγραμμική συνάρτηση των δύο ορισμάτων ή μπορεί να είναι
ένας πίνακας που δίνει μια γραμμικο-ημιγραμμική μορφή.</p></dd><dt><span class="term"><a
name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre class="synopsis">QRDecomposition
(A, Q)</pre><p>Δίνει την ανάλυση QR ενός τετραγωνικού πίνακα <code class="varname">A</code>, επιστρέφει τον
άνω τριγωνικό πίνακα <code class="varname">R</code> και ορίζει το <code class
="varname">Q</code> στον ορθογώνιο (μοναδιαίο) πίνακα. Το <code class="varname">Q</code> πρέπει να είναι μια
αναφορά ή <code class="constant">null</code>, αν δεν θέλετε καμιά επιστροφή. Για παράδειγμα: </p><pre
class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>R =
QRDecomposition(A,&Q)</code></strong>
</pre><p> Θα έχετε τον άνω τριγωνικό πίνακα αποθηκευμένο σε μια μεταβλητή που λέγεται <code
class="varname">R</code> και τον ορθογώνιο (μοναδιαίο) πίνακα αποθηκευμένο στο <code
class="varname">Q</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Επιστρέφει το πηλίκο Ρέιλι (λέγεται επίσης πηλίκο ή λόγος
Ρέιλι-Ρίτζ) ενός πίνακα και ενός διανύσματος.</p><p>
@@ -131,28 +131,37 @@
</p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span></dt><dd><pre
class="synopsis">Rank (M)</pre><p>Παραλλαγές: <code class="function">rank</code></p><p>Δίνει την τάξη ενός
πίνακα.</p><p>
See
<a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
- </p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Επιστρέφει τον πίνακα Ρόσερ, που είναι ένα κλασικό συμμετρικό πρόβλημα δοκιμής
ιδιοτιμής.</p></dd><dt><span class="term"><a
name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Επιστρέφει τον πίνακα που
αντιστοιχεί στην περιστροφή γύρω από το αρχικό στο R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Επιστρέφει τον πίνακα που αντιστοιχεί στην περιστροφή γύρω από τον αρχικό στο R<sup>3</sup> �
�ύρω από τον άξονα x.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre class="synopsis">Rotation3DY
(angle)</pre><p>Επιστρέφει τον πίνακα που αντιστοιχεί στην περιστροφή γύρω από τον αρχικό στο R<sup>3</sup>
γύρω από τον άξονα y.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Επιστρέφει τον πίνακα που αντιστοιχεί στην περιστροφή γύρω από τον αρχικό στο R<sup>3</sup>
γύρω από τον άξονα z.</p></dd><dt><span class="term"><a
name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre class="synopsis">RowSpace (M)</pre><p>Δίνει
έναν πίνακα βάσης για χώρο γραμμών ενός πίνακα.</p></dd><dt><span class="term"><a nam
e="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre class="synopsis">SesquilinearForm
(v,A,w)</pre><p>Υπολογίζει το (v,w) ως προς τη γραμμικο-ημιγραμμική μορφή που δίνεται από τον πίνακα
Α.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Επιστρέφει μια συνάρτηση που υπολογίζει δύο διανύσματα
ως προς τη γραμμικο-ημιγραμμική μορφή που δίνεται από το Α.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Επιστρέφει την κανονική μορφή Σμιθ ενός πίνακα για πεδία
(θα τελειώνει με 1 στη δ
ιαγώνιο).</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Επιλύει το γραμμικό σύστημα Mx=V, επιστρέφει τη λύση
V αν υπάρχει μια μοναδική λύση, αλλιώς <code class="constant">null</code>. Δύο πρ
όσθετες παράμετροι αναφοράς μπορούν να χρησιμοποιηθούν προαιρετικά για να δώσουν τα ανηγμένα M και
V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Επιστρέφει τον πίνακα Toeplitz που κατασκευάστηκε με δεδομένη την πρώτη στήλη c και
(προαιρετικά) την πρώτη γραμμή r. Αν δίνεται μόνο η στήλη c, τότε είναι συζυγής και η μη συζυγής έκδοση
χρησιμοποιείται για να δώσει η πρώτη γραμμή τον ερμιτιανό πίνακα (αν το πρώτο στοιχείο είναι πραγματικός
φυσικά).</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Επιστρέφει τον πίνακα Ρόσερ, που είναι ένα κλασικό συμμετρικό πρόβλημα δοκιμής
ιδιοτιμής.</p></dd><dt><span class="term"><a
name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Επιστρέφει τον πίνακα που
αντιστοιχεί στην περιστροφή γύρω από το αρχικό στο R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Επιστρέφει τον πίνακα που αντιστοιχεί στην περιστροφή γύρω από τον αρχικό στο R<sup>3</sup> �
�ύρω από τον άξονα x.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre class="synopsis">Rotation3DY
(angle)</pre><p>Επιστρέφει τον πίνακα που αντιστοιχεί στην περιστροφή γύρω από τον αρχικό στο R<sup>3</sup>
γύρω από τον άξονα y.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Επιστρέφει τον πίνακα που αντιστοιχεί στην περιστροφή γύρω από τον αρχικό στο R<sup>3</sup>
γύρω από τον άξονα z.</p></dd><dt><span class="term"><a
name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre class="synopsis">RowSpace (M)</pre><p>Δίνει
έναν πίνακα βάσης για χώρο γραμμών ενός πίνακα.</p></dd><dt><span class="term"><a nam
e="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre class="synopsis">SesquilinearForm
(v,A,w)</pre><p>Υπολογίζει το (v,w) ως προς τη γραμμικο-ημιγραμμική μορφή που δίνεται από τον πίνακα
Α.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Επιστρέφει μια συνάρτηση που υπολογίζει δύο διανύσματα
ως προς τη γραμμικο-ημιγραμμική μορφή που δίνεται από το Α.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Επιστρέφει την κανονική μορφή Σμιθ ενός πίνακα για πεδία
(θα τελειώνει με 1 στη δ
ιαγώνιο).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Επιλύει το γραμμικό σύστημα Mx=V, επιστρέφει τη λύση
V αν υπάρχει μια μοναδική λύση, αλλιώς <code class="constant">null</code>. Δύο πρόσθετες παράμετροι αναφοράς
μπορούν να χρησιμοποιηθούν προαιρετικά για να δώσουν τα ανηγμένα M και V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Επιστρέφει τον πίνακα Toeplitz που κατασκευάστηκε με δεδομένη την πρώτη στήλη c και
(προαιρετικά) την πρώτη γραμμή r. Αν δίνεται μόνο η στήλη c, τ�
�τε είναι συζυγής και η μη συζυγής έκδοση χρησιμοποιείται για να δώσει η πρώτη γραμμή τον ερμιτιανό πίνακα
(αν το πρώτο στοιχείο είναι πραγματικός φυσικά).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Παραλλαγές: <code class="function">trace</code></p><p>Υπολογίζει το ίχνος
ενός πίνακα. Δηλαδή, το άθροισμα των διαγώνιων στοιχείων.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Ανάστροφος ενός πίνακα. Αυτός είναι ο ίδιος με τον τελεστή <strong
class="userinput"><code>.'</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
- </p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Παραλλαγές: <code
class="function">vander</code></p><p>Επιστρέφει τον πίνακα Vandermonde.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Vandermonde_matrix"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>Η γωνία των δύο διανυσμάτων ως προς το εσωτερικό γινόμενο που δίνει ο <code
class="varname">B</code>. Αν ο <code class="varname">B</code> δεν δίνεται, τότε το τυπικό ερμιτιανό γινόμενο
χρησιμοποιείται. Ο <code class="varname">B</code> μπορεί είτε ν�
� είναι γραμμικο-ημιγραμμική συνάρτηση δύο ορισμάτων ή μπορεί να είναι ένας πίνακας που δίνει μια
γραμμικο-ημιγραμμική μορφή.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Το ευθύ άθροισμα των διαστημάτων διανύσματος Μ και
Ν.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Τομή των υποχώρων που δίνονται από Μ και
Ν.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>Το άθροισμα των διανυσματικών �
�ώρων M και N, δηλαδή {w | w=m+n, m στο M, n στο N}.</p></dd><dt><span class="term"><a
name="gel-function-adj"></a>adj</span></dt><dd><pre class="synopsis">adj (m)</pre><p>Παραλλαγές: <code
class="function">Adjugate</code></p><p>Δίνει τον κλασικό συζυγή ενός πίνακα.</p></dd><dt><span
class="term"><a name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref
(M)</pre><p>Παραλλαγές: <code class="function">CREF</code><code
class="function">ColumnReducedEchelonForm</code></p><p>Υπολογίζει την ανηγμένη κλιμακωτή μορφή κατά
στήλες.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Παραλλαγές: <code class="function">Determinant</code></p><p>Δίνει την
ορίζουσα ενός πίνακα.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Παραλλαγές: <code
class="function">vander</code></p><p>Επιστρέφει τον πίνακα Vandermonde.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>Η γωνία των δύο διανυσμάτων ως προς το εσωτερικό γινόμενο που δίνει ο <code
class="varname">B</code>. Αν ο <code class="varname">B</code> δεν δίνεται, τότε το τυπικό ερμιτιανό γινόμενο
χρησιμοποιείται. Ο <code class="varname">B</code> μπορεί είτε να είναι γραμμικο-ημιγραμμική συνάρτηση δύο
ορισμάτων ή μπορεί να είναι ένας πίνακας που δίνει μια γραμμικο-ημιγραμμική μορφή.</p></dd><dt><span
class="term"><a name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Το ευθύ άθροισμα των διαστη
μάτων διανύσματος Μ και Ν.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Τομή των υποχώρων που δίνονται από Μ και
Ν.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>Το άθροισμα των διανυσματικών χώρων M και N, δηλαδή {w |
w=m+n, m στο M, n στο N}.</p></dd><dt><span class="term"><a
name="gel-function-adj"></a>adj</span></dt><dd><pre class="synopsis">adj (m)</pre><p>Παραλλαγές: <code
class="function">Adjugate</code></p><p>Δίνει τον κλασικό συζυγή ενός πίνακα.</p></dd><dt><span
class="term"><a name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Πα
ραλλαγές: <code class="function">CREF</code><code
class="function">ColumnReducedEchelonForm</code></p><p>Υπολογίζει την ανηγμένη κλιμακωτή μορφή κατά
στήλες.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Παραλλαγές: <code class="function">Determinant</code></p><p>Δίνει την
ορίζουσα ενός πίνακα.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Παραλλαγές: <code class="function">REF</code><code
class="function">RowEchelonForm</code></p><p>Δίνει την μορφή κλιμακωτής γραμμής ενός πίνακα. Δηλαδή,
εφαρμόζει την απαλοιφή Γκάους, αλλά όχι την πίσω πρόσθεση στο <code class="varname">M</code>. Οι οδηγούσες
γραμμές διαιρούνται για να κάνουν όλους τους οδηγούς 1.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Παραλλαγές: <code class="function">RREF</code><code
class="function">ReducedRowEchelonForm</code></p><p>Δίνει τη ανηγμένη κλιμακωτή μορφή κατά γραμμές ενός
πίνακα. Δηλαδή, εφαρμόζει την απαλοιφή Γκάους μαζί με την πίσω πρόσθεση στο <code
class="varname">M</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
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header"><tr><th colspan="3" align="center">Συνδυαστική Ανάλυση</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s09.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL</th><td widt
h="20%" align="right"> <a accesskey="n" href="ch11s11.html">Επόμενο</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-combinatorics"></a>Συνδυαστική Ανάλυση</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Δίνει τον
<code class="varname">n</code>στό αριθμό Catalan.</p><p>
See
<a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Δίνει όλους τους συνδυασμούς των k αριθμών από 1 μέχρι n ως ένα διάνυσμα διανυσμάτων. (Δείτε
επίσης <a class="link"
href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Διπλό παραγοντικό: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Δίνει όλους τους συνδυασμούς των k αριθμών από 1 μέχρι n ως ένα διάνυσμα διανυσμάτων. (Δείτε
επίσης <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Διπλό παραγοντικό: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
See
<a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Παραγοντικό: <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
@@ -12,13 +15,36 @@
<a class="ulink" href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre
class="synopsis">Fibonacci (x)</pre><p>Παραλλαγές: <code class="function">fib</code></p><p>Υπολογίζει τον
<code class="varname">n</code>στό αριθμό Φιμπονάτσι. Δηλαδή, τον αριθμό που ορίζεται αναδρομικά από <strong
class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</code></strong> και <strong
class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>Υπολογίζει τον αριθμό Φρομπένιους. Δηλαδή, υπολογίζει τον ελάχιστο αριθμό που δεν μπορεί
να δοθεί ως μη αρνητικός ακέραιος γραμμικός συνδυασμός του δοσμένου διανύσματος μη αρνητικών ακεραίων. Το
διάνυσμα μπορεί να δοθεί ως διακριτοί αριθμοί ενός μοναδικού διανύσματος. Όλοι οι δεδομένοι αριθμοί πρέπει να
έχουν ΜΚΔ 1.</p><p>Δείτε <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatri
x"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix (combining_rule)</pre><p>Galois matrix
given a linear combining rule (a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Βρίσκει το διάνυσμα <code class="varname">c</code> μη αρνητικών ακεραίων έτσι ώστε να παίρνει
το εσωτερικό γινόμενο με το <code class="varname">v</code> να είναι ίσο με n. Αν δεν είναι δυνατό, επιστρέφει
<code class="constant">null</code>. Το <code class="varname">v</code> πρέπει να δίνεται ταξινομημένο με
αύξουσα διάταξη και πρέπει να αποτελείται από μη αρνητικούς ακέραιους.</p><p>Δείτε <a class="ulink"
href="http://mathworld.wolfram.com/GreedyAlg
orithm.html" target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Παραλλαγές: <code class="function">HarmonicH</code></p><p>Αρμονικός αριθμός, ο <code
class="varname">n</code>στός αρμονικός αριθμός της τάξης <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Η συνάρτηση Χόφσταντερ q(n) ορίζεται από q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Υπολογίζει τη γραμμική κ
υκλική ακολουθία χρησιμοποιώντας το βηματισμό Γκαλουά.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Υπολογίζει τους πολυωνυμικούς συντελεστές. Παίρνει ένα διάνυσμα από <code
class="varname">k</code> μη αρνητικούς ακέραιους και υπολογίζει τον πολυωνυμικό συντελεστή. Αυτός αντιστοιχεί
με τον συντελεστή στο ομογενές πολυώνυμο σε <code class="varname">k</code> μεταβλητές με τις αντίστοιχες
δυνάμεις.</p><p>Ο τύπος για <strong class="userinput"><code>Multinomial(a,b,c)</code></strong> μπορεί να
γραφτεί ως: </p><pre class="programlisting">(a+b+c)! / (a!b!c!)
+ </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
+ Calculate the Frobenius number. That is calculate largest
+ number that cannot be given as a non-negative integer linear
+ combination of a given vector of non-negative integers.
+ The vector can be given as separate numbers or a single vector.
+ All the numbers given should have GCD of 1.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(combining_rule)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Βρίσκει το διάνυσμα <code class="varname">c</code> μη αρνητικών ακεραίων έτσι ώστε να παίρνει
το εσωτερικό γινόμενο με το <code class="varname">v</code> να είναι ίσο με n. Αν δεν είναι δυνατό, επιστρέφει
<code class="constant">null</code>. Το <code class="varname">v</code> πρέπει να δίνεται ταξινομημένο με
αύξουσα διάταξη και πρέπει να αποτελείται από μη αρνητικούς ακέραιους.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Παραλλαγές: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter (n)</pre><p>Η
συνάρτηση Χόφσταντερ q(n) ορίζεται από q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Υπολογίζει τη γραμμική
κυκλική ακολουθία χρησιμοποιώντας το βηματισμό Γκαλουά.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Υπολογίζει τους πολυωνυμικούς συντελεστές. Παίρνει ένα διάνυσμα από <code
class="varname">k</code> μη αρνητικούς ακέραιους και υπολογίζει τον πολυωνυμικό συντελεστή. Αυτός αντιστοιχεί
με τον συντελεστή στο ομογενές πολυώνυμο σε <code class="varname">k</code> μεταβλητές με τις αντίστοιχ�
�ς δυνάμεις.</p><p>Ο τύπος για <strong class="userinput"><code>Multinomial(a,b,c)</code></strong> μπορεί να
γραφτεί ως: </p><pre class="programlisting">(a+b+c)! / (a!b!c!)
</pre><p> Με άλλα λόγια, αν μπορούμε να έχουμε δύο μόνο στοιχεία, τότε το <strong
class="userinput"><code>Multinomial(a,b)</code></strong> είναι το ίδιο με το <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> ή <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Δίνει τον συνδυασμό που μπορεί να έρθει μετά το v στην κλήση στους συνδυασμούς της συνάρτησης,
ο πρώτος συνδυασμός πρέπει να είναι <strong class="userinput"><code>[1:k]</code></strong>. Αυτή η συνάρτηση
είναι χρήσιμη, αν έχετε πολλούς συνδυασμούς να περάσετε και δεν θέλετε να σπαταλήσετε μνήμη για να τους
αποθηκεύσετε όλους.</p><p>
@@ -34,10 +60,17 @@ do (
) while not IsNull(n:=NextCombination(n,6));</code></strong>
</pre><p>
See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Δίνει το τρίγωνο του Πασκάλ ως πίνακα. Αυτό θα επιστρέψει έναν <code
class="varname">i</code>+1 επί <code class="varname">i</code>+1 κάτω διαγώνιο πίνακα που είναι το τρίγωνο
Πασκάλ μετά από <code class="varname">i</code> επαναλήψεις.</p><p>
See
<a class="ulink" href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Δίνει όλες τις μεταθέσεις των <code class="varname">k</code> αριθμών από 1 μέχρι <code
class="varname">n</code> ως ένα διάνυσμα διανυσμάτων.</p><p>Δείτε <a class="ulink"
href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> ή <a class="ulink"
href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Παραλλαγές: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k
= n(n+1)...(n+(k-1)).</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Δίνει όλες τις μεταθέσεις των <code class="varname">k</code> αριθμών από 1 μέχρι <code
class="varname">n</code> ως ένα διάνυσμα διανυσμάτων.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Παραλλαγές: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k
= n(n+1)...(n+(k-1)).</p><p>
See
<a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Παραλλαγές: <code
class="function">StirlingS1</code></p><p>Ο αριθμός Στέρλινγκ πρώτου είδους.</p><p>
@@ -55,4 +88,8 @@ do (
See
<a class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre
class="synopsis">nPr (n,r)</pre><p>Calculate the number of permutations of size
- <code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>Δείτε <a
class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> ή <a
class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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+ <code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s11.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Γραμμική Άλγεβρα
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diff --git a/help/el/html/ch11s11.html b/help/el/html/ch11s11.html
index 78db037..f2fb421 100644
--- a/help/el/html/ch11s11.html
+++ b/help/el/html/ch11s11.html
@@ -7,7 +7,35 @@
</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Προσπαθεί να υπολογίσει την παράγωγου δοκιμάζοντας πρώτα συμβολικά και έπειτα
αριθμητικά.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Επιστρέφει μια συνάρτηση που είναι άρτια περιοδική
επέκταση της <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>. Δηλαδή μια
συνάρτηση που ορίστηκε στο διάστημα <strong class="userinput"><code>[0,L]</code></strong> επεκτάθηκε για να
είναι άρτια στο <strong class="userinput"><code>[-L,L]</code></strong> και έπειτα επεκτάθηκε να είναι
περιοδική με περίοδο <strong class="userinput"><code>2*L</code></strong>.</p><p>Δείτε επίσης <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> και <a class="link"
href="ch11s11.html#gel-function-Periodi
cExtension">PeriodicExtension</a>.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Επιστρέφει μια συνάρτηση που είναι μια σειρά Φουριέ με
τους συντελεστές δοσμένους από τα διανύσματα <code class="varname">a</code> (ημίτονα) and <code
class="varname">b</code> (συνημίτονα). Σημειώστε ότι, το <strong
class="userinput"><code>a@(1)</code></strong> είναι συντελεστής σταθεράς! Δηλαδή, το <strong
class="userinput"><code>a@(n)</code></strong> αναφέρεται στον όρο <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, ενώ το <strong
class="userinput"><code>b@(n)</code></strong> αναφέρεται στον όρο <strong class="userinput"><code>sin(x*
n*pi/L)</code></strong>. Είτε το <code class="varname">a</code> είτε το <code class="varname">b</code>
μπορεί να είναι <code class="constant">null</code>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο γινόμενο για μια συνάρτηση απλής
παραμέτρου.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Προσπαθεί να υπολογίσει �
�να άπειρο γινόμενο για μια συνάρτηση διπλής παραμέτρου με func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο άθροισμα για μια συνάρτηση απλής
παραμέτρου.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre class="synopsis">InfiniteSum2
(func,arg,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο άθροισμα για μια συνάρτηση διπλής παραμέτρου
με func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Δοκιμάστε για να δείτε αν μια συνάρτηση πραγμα
τικών τιμών είναι συνεχής στο x0 υπολογίζοντας το όριο εκεί.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Δοκιμή διαφορισιμότητας προσεγγίζοντας το αριστερό και δεξιό
όριο και συγκρίνοντας.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Υπολογίζει το αριστερό όριο μιας συνάρτησης πραγματικών στο x0.</p></dd><dt><span
class="term"><a name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit
(f,x0)</pre><p>Υπολογίζει το όριο μιας συνάρτησης πραγματικών στο x0. Προσπαθεί να υπολογίσει και το αριστερό
κα�
� το δεξιό όριο.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Ολοκλήρωση με τον κανόνα μέσου.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Παραλλαγές: <code
class="function">NDerivative</code></p><p>Προσπάθεια υπολογισμού αριθμητικής παραγώγου.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Επιστρέφει μια συνάρτηση που είναι άρτια περιοδική
επέκταση της <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>. Δηλαδή μια
συνάρτηση που ορίστηκε στο διάστημα <strong class="userinput"><code>[0,L]</code></strong> επεκτάθηκε για να
είναι άρτια στο <strong class="userinput"><code>[-L,L]</code></strong> και έπειτα επεκτάθηκε να είναι
περιοδική με περίοδο <strong class="userinput"><code>2*L</code></strong>.</p><p>Δείτε επίσης <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> και <a class="link"
href="ch11s11.html#gel-function-Periodi
cExtension">PeriodicExtension</a>.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Επιστρέφει μια συνάρτηση που είναι μια σειρά Φουριέ με
τους συντελεστές δοσμένους από τα διανύσματα <code class="varname">a</code> (ημίτονα) and <code
class="varname">b</code> (συνημίτονα). Σημειώστε ότι, το <strong
class="userinput"><code>a@(1)</code></strong> είναι συντελεστής σταθεράς! Δηλαδή, το <strong
class="userinput"><code>a@(n)</code></strong> αναφέρεται στον όρο <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, ενώ το <strong
class="userinput"><code>b@(n)</code></strong> αναφέρεται στον όρο <strong class="userinput"><code>sin(x*
n*pi/L)</code></strong>. Είτε το <code class="varname">a</code> είτε το <code class="varname">b</code>
μπορεί να είναι <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο γινόμενο για μια συνάρτηση απλής
παραμέτρου.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο γινόμενο
για μια συνάρτηση διπλής παραμέτρου με func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο άθροισμα για μια συνάρτηση απλής
παραμέτρου.</p></dd><dt><spa
n class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,inc)</pre><p>Προσπαθεί να υπολογίσει ένα άπειρο άθροισμα για
μια συνάρτηση διπλής παραμέτρου με func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Δοκιμάστε για να δείτε αν μια συνάρτηση πραγματικών τιμών είναι συνεχής στο x0 υπολογίζοντας
το όριο εκεί.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Δοκιμή διαφορισιμότητας προσεγγίζοντας το αριστερό και δεξιό
όριο και συγκρίνοντας.</p></dd><dt><s
pan class="term"><a name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre
class="synopsis">LeftLimit (f,x0)</pre><p>Υπολογίζει το αριστερό όριο μιας συνάρτησης πραγματικών στο
x0.</p></dd><dt><span class="term"><a name="gel-function-Limit"></a>Limit</span></dt><dd><pre
class="synopsis">Limit (f,x0)</pre><p>Υπολογίζει το όριο μιας συνάρτησης πραγματικών στο x0. Προσπαθεί να
υπολογίσει και το αριστερό και το δεξιό όριο.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Ολοκλήρωση με τον κανόνα μέσου.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Παραλλαγές: <code class
="function">NDerivative</code></p><p>Προσπάθεια υπολογισμού αριθμητικής παραγώγου.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Επιστρέφει ένα διάνυσμα διανυσμάτων
<strong class="userinput"><code>[a,b]</code></strong> όπου το <code class="varname">a</code> είναι οι
συντελεστές συνημιτόνου και το <code class="varname">b</code> είναι οι συντελεστές ημιτόνου της σειράς Φουριέ
του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code> (που ορίζεται στο <strong
class="userinput"><code>[-L,L]</code></strong> και επεκτείνεται περιοδικά) με συντελεστές μέχρι τον <code
class="varname">N</code>στό αρμονικό που υπολογίζεται αριθμητικά. Οι συν�
�ελεστές υπολογίζονται με αριθμητική ολοκλήρωση χρησιμοποιώντας το <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Επιστρέφει μια συνάρτηση που είναι η σειρά
Φουριέ του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code> (που ορίζεται στο
<strong class="use
rinput"><code>[-L,L]</code></strong> και επεκτείνεται περιοδικά) με συντελεστές μέχρι τον <code
class="varname">N</code>στό αρμονικό που υπολογίζεται αριθμητικά. Αυτή είναι η τριγωνομετρική πραγματική
σειρά που αποτελείται από ημίτονα και συνημίτονα. Οι συντελεστές υπολογίζονται με αριθμητική ολοκλήρωση
χρησιμοποιώντας το <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p><p>Version 1.0.7 onwards.</p></dd><dt><span clas
s="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Επιστρέφει ένα διάνυσμα
συντελεστών της σειράς Φουριέ συνημιτόνου του <code class="function">f</code> με ημιπερίοδο <code
class="varname">L</code>. Δηλαδή, η <code class="function">f</code> ορισμένη στο <strong
class="userinput"><code>[0,L]</code></strong> παίρνει την άρτια περιοδική επέκταση και υπολογίζει τη σειρά
Φουριέ, η οποία έχει μόνο όρους συνημιτόνου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό
αρμονικό. Οι συντελεστές υπολογίζονται με αριθμητική ολοκλήρωση χρησιμοποι�
�ντας την <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>. Σημειώστε ότι το <strong
class="userinput"><code>a@(1)</code></strong> είναι ο συντελεστής σταθεράς! Δηλαδή, <strong
class="userinput"><code>a@(n)</code></strong> αναφέρεται στον όρο <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> Για περισσότερες
πληροφορίες.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Επιστ�
�έφει μια συνάρτηση που είναι η σειρά Φουριέ συνημιτόνου του <code class="function">f</code> με ημιπερίοδο
<code class="varname">L</code>. Δηλαδή, παίρνουμε την <code class="function">f</code> ορισμένη στο <strong
class="userinput"><code>[0,L]</code></strong>, παίρνει την άρτια περιοδική επέκταση και υπολογίζει τη σειρά
Φουριέ, η οποία έχει μόνο όρους συνημιτόνου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό
αρμονικό. Οι συντελεστές υπολογίζονται με αριθμητική ολοκλήρωση χρησιμοποιώντας την <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fo
urier_series" target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> Για περισσότερες
πληροφορίες.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Επιστρέφει ένα διάνυσμα συντελεστών
της σειράς Φουριέ ημιτόνου του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>.
Δηλαδή, η <code class="function">f</code> ορισμένη στο <strong class="userinput"><code>[0,L]</code></strong>
παίρνει την περιττή περιοδική επέκταση και υπολογίζει τη σειρά Φουριέ, η οποία έχει μόνο όρους ημιτ�
�νου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό αρμονικό. Οι συντελεστές
υπολογίζονται με αριθμητική ολοκλήρωση χρησιμοποιώντας την <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Επιστρέφει μια συνάρτηση που είναι η
σειρά Φουρ
ιέ ημιτόνου του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>. Δηλαδή,
παίρνουμε τη <code class="function">f</code> ορισμένη στο <strong
class="userinput"><code>[0,L]</code></strong>, παίρνει την άρτια περιοδική επέκταση και υπολογίζει τη σειρά
Φουριέ, η οποία έχει μόνο όρους ημιτόνου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό
αρμονικό. Οι συντελεστές υπολογίζονται με αριθμητική ολοκλήρωση χρησιμοποιώντας την <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> ή <a class="ulink"
href="http://mathworld.wolfr
am.com/FourierSineSeries.html" target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.7
onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Ολοκλήρωση με τον κανόνα που ορίστηκε στο
NumericalIntegralFunction του f από το a μέχρι το b χρησιμοποιώντας βήματα
NumericalIntegralSteps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Προσπαθεί να υπολογίσει την αριθμητική αριστερή
παράγωγο.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step
_fun,tolerance,successive_for_success,N)</pre><p>Προσπαθεί να υπολογίσει το όριο του f(step_fun(i)) καθώς το
i πηγαίνει από 1 έως N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Προσπαθεί να υπολογίσει την αριθμητική δεξιά
παράγωγο.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Επιστρέφει μια συνάρτηση που είναι περιττή περιοδική
επέκταση της <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>. Δηλαδή μια
συνάρτηση που ορίστηκε στο διάστημα <strong class="userinput"><code>[0,L]</code></
strong> επεκτάθηκε για να είναι περιττή στο <strong class="userinput"><code>[-L,L]</code></strong> και
έπειτα επεκτάθηκε να είναι περιοδική με περίοδο <strong
class="userinput"><code>2*L</code></strong>.</p><p>Δείτε επίσης <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> και <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Version 1.0.7
onwards.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Υπολογίζει τη μονόπλευρη παράγωγο χρησιμοποιώντας
τον τύπο πέντε σημείων.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre cla
ss="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Υπολογίζει τη μονόπλευρη παράγωγο χρησιμοποιώντας
τον τύπο τριών σημείων.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Επιστρέφει μια συνάρτηση που είναι η περιοδική επέκταση
της <code class="function">f</code> ορισμένη στο διάστημα <strong
class="userinput"><code>[a,b]</code></strong> και έχει περίοδο <strong
class="userinput"><code>b-a</code></strong>.</p><p>Δείτε επίσης <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> και <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>Version 1.0.7
onwards.</p></dd><dt><span class="term"><a name="gel-func
tion-RightLimit"></a>RightLimit</span></dt><dd><pre class="synopsis">RightLimit (f,x0)</pre><p>Υπολογίζει το
δεξιό όριο μιας συνάρτησης πραγματικών στο x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Υπολογίζει τη δίπλευρη παράγωγο χρησιμοποιώντας
τον τύπο πέντε σημείων.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Υπολογίζει τη δίπλευρη παράγωγο χρησιμοποιώντας
τον τύπο τριών σημείων.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
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+ </p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Επιστρέφει ένα διάνυσμα διανυσμάτων
<strong class="userinput"><code>[a,b]</code></strong> όπου το <code class="varname">a</code> είναι οι
συντελεστές συνημιτόνου και το <code class="varname">b</code> είναι οι συντελεστές ημιτόνου της σειράς Φουριέ
του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code> (που ορίζεται στο <strong
class="userinput"><code>[-L,L]</code></strong> και επεκτείνεται περιοδικά) με συντελεστές μέχρι τον <code
class="varname">N</code>στό αρμονικό που υπολογίζεται αριθμητικά. Οι συν�
�ελεστές υπολογίζονται με αριθμητική ολοκλήρωση χρησιμοποιώντας το <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Επιστρέφει μια συνάρτηση που είναι η σειρά
Φουριέ του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code> (που ορίζεται στο
<strong class="userinput"><code>[-L,L]</code></strong> και επεκτείνεται περιοδικά) με συντελεστές μέχρι τον
<code class="varname">N</code>στό αρμονικό που υπολογίζεται αριθμητικά. Αυτή είναι η τριγωνομετρική
πραγματική σειρά που αποτελείται από ημίτονα και συνημίτονα. Οι συντελεστές υπολογίζονται με αριθμητική
ολοκλήρωσ
η χρησιμοποιώντας το <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Επιστρέφει ένα διάνυσμα
συντελεστών της σειράς Φουριέ συνημιτόνου του <code class="function">f</code> με ημιπερίοδο <code
class="varname">L</code>. Δηλαδή, η <code class="function">f</code> ορισμένη στο <strong
class="userinput"><code>[0,L]</code></strong> παίρνει την άρτια περιοδική επέκταση και υπολογίζει τη σειρά
Φουριέ, η οποία έχει μόνο όρους συνημιτόνου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό
αρμονικό. Οι συντελεστές υπολογίζονται με �
�ριθμητική ολοκλήρωση χρησιμοποιώντας την <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.
Σημειώστε ότι το <strong class="userinput"><code>a@(1)</code></strong> είναι ο συντελεστής σταθεράς! Δηλαδή,
<strong class="userinput"><code>a@(n)</code></strong> αναφέρεται στον όρο <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Επιστρέφει μια συνάρτηση που είναι η
σειρά Φουριέ συνημιτόνου του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>.
Δηλαδή, παίρνουμε την <code class="function">f</code> ορισμένη στο <strong
class="userinput"><code>[0,L]</code></strong>, παίρνει την άρτια περιοδική επέκταση και υπολογίζει τη σειρά
Φουριέ, η οποία έχει μόνο όρους συνημιτόνου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό
αρμονικό. Οι συντελεστές υπολογίζονται μ�
� αριθμητική ολοκλήρωση χρησιμοποιώντας την <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Επιστρέφει ένα διάνυσμα συντελεστών
της σειράς Φουριέ ημιτόνου του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>.
Δηλαδή, η <code class="function">f</code> ορισμένη στο <strong class="userinput"><code>[0,L]</code></strong>
παίρνει την περιττή περιοδική επέκταση και υπολογίζει τη σειρά Φουριέ, η οποία έχει μόνο όρους ημιτόνου. Η
σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό αρμονικό. Οι συντελεστές υπολογίζονται με
αριθμητ�
�κή ολοκλήρωση χρησιμοποιώντας την <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Επιστρέφει μια συνάρτηση που είναι η
σειρά Φουριέ ημιτόνου του <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>.
Δηλαδή, παίρνουμε τη <code class="function">f</code> ορισμένη στο <strong
class="userinput"><code>[0,L]</code></strong>, παίρνει την άρτια περιοδική επέκταση και υπολογίζει τη σειρά
Φουριέ, η οποία έχει μόνο όρους ημιτόνου. Η σειρά υπολογίζεται μέχρι τον <code class="varname">N</code>στό
αρμονικό. Οι συντελεστές υπολογίζονται με αριθμητικ
ή ολοκλήρωση χρησιμοποιώντας την <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Ολοκλήρωση με τον κανόνα που ορίστηκε στο
NumericalIntegralFunction του f από το a μέχρι το b χρησιμοποιώντας βήματα
NumericalIntegralSteps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Προσπαθεί να υπολογίσει την αριθμητική αριστερή
παράγωγο.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Προσπαθεί
να υπολογίσει το όρ�
�ο του f(step_fun(i)) καθώς το i πηγαίνει από 1 έως N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Προσπαθεί να υπολογίσει την αριθμητική δεξιά
παράγωγο.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Επιστρέφει μια συνάρτηση που είναι περιττή περιοδική
επέκταση της <code class="function">f</code> με ημιπερίοδο <code class="varname">L</code>. Δηλαδή μια
συνάρτηση που ορίστηκε στο διάστημα <strong class="userinput"><code>[0,L]</code></strong> επεκτάθηκε για να
είναι περιττή στο <strong class="userinput"><cod
e>[-L,L]</code></strong> και έπειτα επεκτάθηκε να είναι περιοδική με περίοδο <strong
class="userinput"><code>2*L</code></strong>.</p><p>Δείτε επίσης <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> και <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Version 1.0.7
onwards.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Υπολογίζει τη μονόπλευρη παράγωγο χρησιμοποιώντας
τον τύπο πέντε σημείων.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Υπολογίζει τη μονόπλευρη
παράγωγο χρησιμοποιώντας τον τύπο τριών σημείων.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Επιστρέφει μια συνάρτηση που είναι η περιοδική επέκταση
της <code class="function">f</code> ορισμένη στο διάστημα <strong
class="userinput"><code>[a,b]</code></strong> και έχει περίοδο <strong
class="userinput"><code>b-a</code></strong>.</p><p>Δείτε επίσης <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> και <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>Version 1.0.7
onwards.</p></dd><dt><span class="term"><a name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre
class="synopsis">RightLimit (f,x0)</pre><p>Υπολ�
�γίζει το δεξιό όριο μιας συνάρτησης πραγματικών στο x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Υπολογίζει τη δίπλευρη παράγωγο χρησιμοποιώντας
τον τύπο πέντε σημείων.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Υπολογίζει τη δίπλευρη παράγωγο χρησιμοποιώντας
τον τύπο τριών σημείων.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s10.html">Προηγ</a>
</td><td width="20%" align="center"><a accesskey="u" href
="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s12.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Συνδυαστική Ανάλυση
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td width="40%"
align="right" valign="top"> Συναρτήσεις</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s12.html b/help/el/html/ch11s12.html
index 820dd71..5fdb019 100644
--- a/help/el/html/ch11s12.html
+++ b/help/el/html/ch11s12.html
@@ -1,4 +1,22 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Συναρτήσεις</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html"
title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της GEL"><link rel="prev" href="ch11s11.html" title="Μαθηματική
Ανάλυση"><link rel="next" href="ch11s13.html" title="Επίλυση εξίσωσης"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Συναρτήσεις</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s11.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11.
Κατάλογος συναρτήσεων της GEL</th><td width="20%" align="right"> <a acc
esskey="n" href="ch11s13.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-functions"></a>Συναρτήσεις</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Παραλλαγές: <code class="function">Arg</code><code
class="function">arg</code></p><p>όρισμα (γωνία) μιγαδικού αριθμού.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0 (x)</pre><p>Η
συνάρτηση Μπεσέλ πρώτου είδους τάξης 0. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>Δείτε <a
class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wik
ipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Η
συνάρτηση Μπεσέλ πρώτου είδους τάξης 1. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>Δείτε <a
class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> για
περισσότερες πληροφορίες.</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Η
συνάρτηση Μπεσέλ πρώτου είδους τάξης <code class="varname">n</code>. Εφαρμόζεται μόνο για πραγματικούς
αριθμούς.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_fu
nctions" target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.16
onwards.</p></dd><dt><span class="term"><a name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre
class="synopsis">BesselY0 (x)</pre><p>Η συνάρτηση Μπεσέλ δεύτερου είδους τάξης 0. Εφαρμόζεται μόνο για
πραγματικούς αριθμούς.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.16 onwards.</p></dd><dt><span
class="term"><a name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1
(x)</pre><p>Η συνάρτηση Μπεσέλ δεύτερου είδους τάξης 1. Εφαρμόζεται μόνο για πραγματικούς
αριθμούς.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel
_functions" target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.16
onwards.</p></dd><dt><span class="term"><a name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre
class="synopsis">BesselYn (n,x)</pre><p>Η συνάρτηση Μπεσέλ δεύτερου είδους τάξης <code
class="varname">n</code>. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis
">DiscreteDelta (v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span
class="term"><a name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre
class="synopsis">ErrorFunction (x)</pre><p>Παραλλαγές: <code class="function">erf</code></p><p>Η συνάρτηση
σφάλματος, 2/sqrt(pi) * int_0^x e^(-t^2) dt.</p><p>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Συναρτήσεις</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html"
title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της GEL"><link rel="prev" href="ch11s11.html" title="Μαθηματική
Ανάλυση"><link rel="next" href="ch11s13.html" title="Επίλυση εξίσωσης"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Συναρτήσεις</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s11.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11.
Κατάλογος συναρτήσεων της GEL</th><td width="20%" align="right"> <a acc
esskey="n" href="ch11s13.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-functions"></a>Συναρτήσεις</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Παραλλαγές: <code class="function">Arg</code><code
class="function">arg</code></p><p>όρισμα (γωνία) μιγαδικού αριθμού.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0 (x)</pre><p>Η
συνάρτηση Μπεσέλ πρώτου είδους τάξης 0. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Η
συνάρτηση Μπεσέλ πρώτου είδους τάξης 1. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Η
συνάρτηση Μπεσέλ πρώτου είδους τάξης <code class="varname">n</code>. Εφαρμόζεται μόνο για πραγματικούς
αριθμούς.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Η
συνάρτηση Μπεσέλ δεύτερου είδους τάξης 0. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Η
συνάρτηση Μπεσέλ δεύτερου είδους τάξης 1. Εφαρμόζεται μόνο για πραγματικούς αριθμούς.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Η
συνάρτηση Μπεσέλ δεύτερου είδους τάξης <code class="varname">n</code>. Εφαρμόζεται μόνο για πραγματικούς
αριθμούς.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Παραλλαγές: <code class="function">erf</code></p><p>Η συνάρτηση σφάλματος, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ErrorFunction"; target="_top">Planetmath</a> for more
information.
@@ -8,24 +26,41 @@
</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Παραλλαγές: <code class="function">Gamma</code></p><p>Η συνάρτηση γάμα. Προς το παρόν υλοποιείται
μόνο για πραγματικούς.</p><p>
See
<a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Επιστρέφει 1 αν και μόνο αν όλα τα στοιχεία είναι ίσα.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>Ο βασικός
κλάδος της συνάρτησης W Λαμπέρ υπολογίζεται μόνο για πραγματικές τιμές μεγαλύτερες ή ίσες από <strong
class="userinput"><code>-1/e</code></strong>. Δηλαδή, <code class="function">LambertW</code> είναι το
αντίστροφο της παράστασης <strong class="userinput"><code>x*e^x</code></strong>. Ακόμα για πραγματικούς <code
class="varname">x</code> αυτή η παράσταση δεν είναι ένα προς ένα και συνεπώς έχει
δύο κλάδους στο <strong class="userinput"><code>[-1/e,0)</code></strong>. Δείτε <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> για τον άλλο
πραγματικό κλάδο.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.18 onwards.</p></dd><dt><span
class="term"><a name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1
(x)</pre><p>Ο κλάδος μείον-ένα της συνάρτησης W Λαμπέρ υπολογίζεται μόνο για πραγματικές τιμές μεγαλύτερες ή
ίσες με <strong class="userinput"><code>-1/e</code></strong> και μικρότερες από 0. Δηλαδή, το <code
class="function">LambertWm1</code> είναι ο δεύτερος κλάδος
του αντίστροφου <strong class="userinput"><code>x*e^x</code></strong>. Δείτε <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> για τον βασικό
κλάδο.</p><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Βρίσκει την πρώτη τιμή όπου f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Μετασχηματισμός Μέμπιους του δίσκου στον εαυτόν του,
απεικονίζοντας το a στο 0.</p><p>
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Επιστρέφει 1 αν και μόνο αν όλα τα στοιχεία είναι ίσα.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>Ο βασικός
κλάδος της συνάρτησης W Λαμπέρ υπολογίζεται μόνο για πραγματικές τιμές μεγαλύτερες ή ίσες από <strong
class="userinput"><code>-1/e</code></strong>. Δηλαδή, <code class="function">LambertW</code> είναι το
αντίστροφο της παράστασης <strong class="userinput"><code>x*e^x</code></strong>. Ακόμα για πραγματικούς <code
class="varname">x</code> αυτή η παράσταση δεν είναι ένα προς ένα και συνεπώς έχει
δύο κλάδους στο <strong class="userinput"><code>[-1/e,0)</code></strong>. Δείτε <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> για τον άλλο
πραγματικό κλάδο.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>Ο
κλάδος μείον-ένα της συνάρτησης W Λαμπέρ υπολογίζεται μόνο για πραγματικές τιμές μεγαλύτερες ή ίσες με
<strong class="userinput"><code>-1/e</code></strong> και μικρότερες από 0. Δηλαδή, το <code
class="function">LambertWm1</code> είναι ο δεύτερος κλάδος του αντίστροφου <strong
class="userinput"><code>x*e^x</code></strong>. Δείτε <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> για τον βασικό
κλάδο.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Βρίσκει την πρώτη τιμή όπου f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Μετασχηματισμός Μέμπιους του δίσκου στον εαυτόν του,
απεικονίζοντας το a στο 0.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Μετασχηματισμός Μέμπιους χρησιμοποιώντας τον διπλό λόγο παίρνοντας z2,z3,z4 στο 1,0, και
άπειρο αντίστοιχα.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Απεικόνιση Μέμπιους χρησιμοποιώντας τον διπλό
λόγο παίρνοντας άπειρο στο άπειρο και z2,z3 στο 1 και 0 αντίστοιχα.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Μετασχηματισμός Μέμπιους χρησιμοποιώντας τον
διπλό λόγο παίρνοντας άπειρο στο 1 και z3,z4 στο 0 και άπειρο αντίστοιχα.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Μετασχηματισμός Μέμπιους χρησιμοποιώντας τον
διπλό λόγο παίρνοντας άπειρο στο 0 και z2,z4 στο 1 και άπειρο αντίστοιχα.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Πυρήνας Πουασόν στο D(0,1) (μη κανονικοποιημένο στο 1, δηλαδή το ολοκλήρωμα αυτού είναι
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Πυρήνας Πουασόν στο D(0,R) (μη κανονικοποιημένο στο
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Παραλλαγές: <code class="function">zeta</code></p><p>Η συνάρτηση
ζήτα Ρίμαν. Προς το παρόν υλοποιείται μόνο για πραγματικούς.</p><p>
See
<a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
- </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>Το μοναδιαίο βήμα συνάρτησης είναι 0 για x<0, 1 αλλιώς. Αυτό είναι
το ολοκλήρωμα της συνάρτησης δέλτα Ντιράκ. Λέγεται επίσης συνάρτηση Χέβισαϊντ.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>Η συνάρτηση <code class="function">cis</code>, δηλαδή είναι η ίδια με τη
<strong class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><
p>Μετατρέπει βαθμούς σε ακτίνια.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Μετατρέπει
ακτίνια σε μοίρες.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Υπολογίζει τη μη κανονικοποιημένη συνάρτηση sinc, δηλαδή την <strong
class="userinput"><code>sin(x)/x</code></strong>. Αν θέλετε την κανονικοποιημένη συνάρτηση καλέστε <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>Δείτε <a class="ulink"
href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> για περισσότερες
πληροφορίες.</p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width=
"40%" align="left"><a accesskey="p" href="ch11s11.html">Προηγ</a> </td><td width="20%" align="center"><a
accesskey="u" href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Μαθηματική Ανάλυση
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td width="40%"
align="right" valign="top"> Επίλυση εξίσωσης</td></tr></table></div></body></html>
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>Το μοναδιαίο βήμα συνάρτησης είναι 0 για x<0, 1 αλλιώς. Αυτό είναι
το ολοκλήρωμα της συνάρτησης δέλτα Ντιράκ. Λέγεται επίσης συνάρτηση Χέβισαϊντ.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>Η συνάρτηση <code class="function">cis</code>, δηλαδή είναι η ίδια με τη
<strong class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Μετατρέπει
βαθμούς σε ακτίνια.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Μετατρέπει
ακτίνια σε μοίρες.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Υπολογίζει τη μη κανονικοποιημένη συνάρτηση sinc, δηλαδή την <strong
class="userinput"><code>sin(x)/x</code></strong>. Αν θέλετε την κ�
�νονικοποιημένη συνάρτηση καλέστε <strong class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ </p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Προηγ</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Μαθηματική Ανάλυση
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td width="40%"
align="right" valign="top"> Επίλυση εξίσωσης</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s13.html b/help/el/html/ch11s13.html
index 1615e81..868f40a 100644
--- a/help/el/html/ch11s13.html
+++ b/help/el/html/ch11s13.html
@@ -2,7 +2,7 @@
See
<a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
@@ -15,12 +15,12 @@
</p><p>Τα συστήματα μπορούν να επιλυθούν έχοντας απλά το <code class="varname">y</code> να είναι
ένα διάνυσμα (στήλης) παντού. Δηλαδή, το <code class="varname">y0</code> μπορεί να είναι ένα διάνυσμα, οπότε
το <code class="varname">f</code> πρέπει να πάρει έναν αριθμό <code class="varname">x</code> και ένα διάνυσμα
του ίδιου μεγέθους για το δεύτερο όρισμα και πρέπει να επιστρέψει ένα διάνυσμα του ίδιου μεγέθους.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
@@ -53,7 +53,7 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Βρίσκει τις ρίζες μιας συνάρτησης χρησιμοποιώντας τη
μέθοδο διχοτόμησης. Τα <code class="varname">a</code> και <code class="varname">b</code> είναι το αρχικό
διάστημα πρόβλεψης, τα <strong class="userinput"><code>f(a)</code></strong> και <strong
class="userinput"><code>f(b)</code></strong> πρέπει να έχουν αντίθετα πρόσημα. Το <code
class="varname">TOL</code> είναι η επιθυμητή ανοχή και <code class="varname">N</code> είναι το όριο στον
αριθμό των επαναλήψεων εκτέλεσης, 0 σημαίνει χωρίς όριο. Η συνάρτηση επιστρέφει ένα διάνυσμα <str
ong class="userinput"><code>[success,value,iteration]</code></strong>, όπου <code
class="varname">success</code> είναι μια λογική τιμή που δείχνει επιτυχία, <code class="varname">value</code>
είναι η τελευταία υπολογισμένη τιμή και <code class="varname">iteration</code> είναι ο αριθμός των
επαναλήψεων που έγιναν.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Βρίσκει τις ρίζες μιας συνάρτησης
χρησιμοποιώντας τη μέθοδο ψευδούς θέσης. Τα <code class="varname">a</code> και <code class="varname">b</code>
είναι το αρχικό διάστημα πρόβλεψης, τα <strong class="userinput"><code>f(a)</code></strong> και <strong
class="userinput"><code>f(b)<
/code></strong> πρέπει να έχουν αντίθετα πρόσημα. Το <code class="varname">TOL</code> είναι η επιθυμητή
ανοχή και <code class="varname">N</code> είναι το όριο στον αριθμό των επαναλήψεων εκτέλεσης, 0 σημαίνει
χωρίς όριο. Η συνάρτηση επιστρέφει ένα διάνυσμα <strong
class="userinput"><code>[success,value,iteration]</code></strong>, όπου <code class="varname">success</code>
είναι μια λογική τιμή που δείχνει επιτυχία, <code class="varname">value</code> είναι η τελευταία υπολογισμένη
τιμή και <code class="varname">iteration</code> είναι ο αριθμός των επαναλήψεων που έγιναν.</p></dd><dt><span
class="term"><a name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMeth
od (f,x0,x1,x2,TOL,N)</pre><p>Βρίσκει τις ρίζες μιας συνάρτησης χρησιμοποιώντας τη μέθοδο Μίλερ. Το <code
class="varname">TOL</code> είναι η επιθυμητή ανοχή και <code class="varname">N</code> είναι το όριο στον
αριθμό των επαναλήψεων εκτέλεσης, 0 σημαίνει χωρίς όριο. Η συνάρτηση επιστρέφει ένα διάνυσμα <strong
class="userinput"><code>[success,value,iteration]</code></strong>, όπου <code class="varname">success</code>
είναι μια λογική τιμή που δείχνει επιτυχία, <code class="varname">value</code> είναι η τελευταία υπολογισμένη
τιμή και <code class="varname">iteration</code> είναι ο αριθμός των επαναλήψεων που έγιναν.</p></dd><dt><span
class="term"><a name="gel-function-FindRootSecant"></a>FindRootS
ecant</span></dt><dd><pre class="synopsis">FindRootSecant (f,a,b,TOL,N)</pre><p>Βρίσκει τις ρίζες μιας
συνάρτησης χρησιμοποιώντας τη μέθοδο τέμνουσας. Τα <code class="varname">a</code> και <code
class="varname">b</code> είναι το αρχικό διάστημα πρόβλεψης, τα <strong
class="userinput"><code>f(a)</code></strong> και <strong class="userinput"><code>f(b)</code></strong> πρέπει
να έχουν αντίθετα πρόσημα. Το <code class="varname">TOL</code> είναι η επιθυμητή ανοχή και <code
class="varname">N</code> είναι το όριο στον αριθμό των επαναλήψεων εκτέλεσης, 0 σημαίνει χωρίς όριο. Η
συνάρτηση επιστρέφει ένα διάνυσμα <strong class="userinput"><code>[success,value,iteration]</code></strong>,
όπου <code class="varname">success</code> είναι �
�ια λογική τιμή που δείχνει επιτυχία, <code class="varname">value</code> είναι η τελευταία υπολογισμένη τιμή
και <code class="varname">iteration</code> είναι ο αριθμός των επαναλήψεων που έγιναν.</p></dd><dt><span
class="term"><a name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre
class="synopsis">HalleysMethod (f,df,ddf,guess,epsilon,maxn)</pre><p>Find zeros using Halley's method. <code
class="varname">f</code> is
the function, <code class="varname">df</code> is the derivative of
<code class="varname">f</code>, and <code class="varname">ddf</code> is the second
derivative of
@@ -61,27 +61,34 @@
guess. The function returns after two successive values are
within <code class="varname">epsilon</code> of each other, or after <code
class="varname">maxn</code> tries, in which case the function returns <code class="constant">null</code>
indicating failure.
</p><p>Δείτε επίσης <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a> και <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Παράδειγμα εύρεσης της τετραγωνικής ρίζας του 10:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
-</pre><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.18 onwards.</p></dd><dt><span
class="term"><a name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre
class="synopsis">NewtonsMethod (f,df,guess,epsilon,maxn)</pre><p>Βρίσκει μηδενικά χρησιμοποιώντας τη μέθοδο
Νεύτωνα. Το <code class="varname">f</code> είναι η συνάρτηση και <code class="varname">df</code> είναι η
παράγωγος του <code class="varname">f</code>. Η <code class="varname">guess</code> είναι η αρχική πρόβλεψη. Η
συνάρτηση επιστρέφει μετά από δύο διαδοχικές τιμές που είναι μέσα στο <code class="varname">epsilon</code>
μεταξύ τους, ή μετά από <code class="varname">maxn</code> προσ
πάθειες, οπότε η συνάρτηση επιστρέφει <code class="constant">null</code> που δείχνει αποτυχία.</p><p>Δείτε
επίσης <a class="link" href="ch11s15.html#gel-function-NewtonsMethodPoly"><code
class="function">NewtonsMethodPoly</code></a> and <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Παράδειγμα εύρεσης της τετραγωνικής ρίζας του 10:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
-</pre><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p><p>Version 1.0.18 onwards.</p></dd><dt><span
class="term"><a name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre
class="synopsis">PolynomialRoots (p)</pre><p>Υπολογίζει ρίζες ενός πολυωνύμου (βαθμών από 1 μέχρι 4)
χρησιμοποιώντας τους τύπους για τέτοια πολυώνυμα. Το πολυώνυμο πρέπει να δίνεται ως ένα διάνυσμα συντελεστών.
Δηλαδή το <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> αντιστοιχεί στο διάνυσμα <strong
class="userinput"><code>[1,2,0,4]</code></strong>. Επιστρέφει ένα διάνυσμα στήλης των λύσεων.</p><p>Η
συνάρτηση καλεί <a class="link" href="ch
11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> και <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre
class="synopsis">QuadraticFormula (p)</pre><p>Υπολογίζει ρίζες ενός δευτεροβάθμιου πολυωνύμου (βαθμού 2)
χρησιμοποιώντας τον τύπο δευτεροβάθμιας. Το πολυώνυμο πρέπει να δίνεται ως ένα διάνυσμα συντελεστών. Δηλαδή
το <strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> αντιστοιχεί στο διάνυσμα <strong
class="userinput"><code>[1,2,3]</code></strong>. Επιστρέφει ένα διάνυσμα στήλης των δύο λύσεων.</p><p>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Βρίσκει μηδενικά χρησιμοποιώντας τη μέθοδο Νεύτωνα. Το <code
class="varname">f</code> είναι η συνάρτηση και <code class="varname">df</code> είναι η παράγωγος του <code
class="varname">f</code>. Η <code class="varname">guess</code> είναι η αρχική πρόβλεψη. Η συνάρτηση
επιστρέφει μετά από δύο διαδοχικές τιμές που είναι μέσα στο <code class="varname">epsilon</code> μεταξύ τους,
ή μετά από <code class="varname">maxn</code> προσπάθειες, οπότε η συνάρτηση επιστρέφει <code
class="constant">null</code> που δείχνει αποτυχία.</p><p>Δείτε επίσ�
�ς <a class="link" href="ch11s15.html#gel-function-NewtonsMethodPoly"><code
class="function">NewtonsMethodPoly</code></a> and <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Παράδειγμα εύρεσης της τετραγωνικής ρίζας του 10:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Υπολογίζει ρίζες ενός πολυωνύμου (βαθμών από 1 μέχρι 4) χρησιμοποιώντας τους τύπους για τέτοια
πολυώνυμα. Το πολυώνυμο πρέπει να δίνεται ως ένα διάνυσμα συντελεστών. Δηλαδή το <strong
class="userinput"><code>4*x^3 + 2*x + 1</code></strong> αντιστοιχεί στο διάνυσμα <strong
class="userinput"><code>[1,2,0,4]</code></strong>. Επιστρέφει ένα διάνυσμα στήλης των λύσεων.</p><p>Η
συνάρτηση καλεί <a class="link" href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a
class="link" href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> και <a class="link" hr
ef="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre
class="synopsis">QuadraticFormula (p)</pre><p>Υπολογίζει ρίζες ενός δευτεροβάθμιου πολυωνύμου (βαθμού 2)
χρησιμοποιώντας τον τύπο δευτεροβάθμιας. Το πολυώνυμο πρέπει να δίνεται ως ένα διάνυσμα συντελεστών. Δηλαδή
το <strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> αντιστοιχεί στο διάνυσμα <strong
class="userinput"><code>[1,2,3]</code></strong>. Επιστρέφει ένα διάνυσμα στήλης των δύο λύσεων.</p><p>
See
- <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Υπολογίζει ρίζες ενός τεταρτοβάθμιου πολυωνύμου (βαθμού 4) χρησιμοποιώντας τον τύπο
τεταρτοβάθμιας. Το πολυώνυμο πρέπει να δίνεται ως ένα διάνυσμα συντελεστών. Δηλαδή το <strong
class="userinput"><code>5*x^4 + 2*x + 1</code></strong> αντιστοιχεί στο διάνυσμα <strong
class="userinput"><code>[1,2,0,0,5]</code></strong>. Επιστρέφει ένα διάνυσμα στήλης τεσσάρων λύσεων.</p><p>
See
<a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Χρησιμοποιεί την κλασική μη αναπροσαρμοστική μέθοδο τέταρτης τάξης Runge-Kutta για
αριθμητική επίλυση της y'=f(x,y) με αρχικά <code class="varname">x0</code>, <code class="varname">y0</code>
πηγαίνει στο <code class="varname">x1</code> με βήματα <code class="varname">n</code>, επιστρέφει <code
class="varname">y</code> στο <code class="varname">x1</code>.</p><p>Τα συστήματα μπορούν να επιλυθούν έχοντας
απλά το <code class="varname">y</code> να είναι ένα διάνυσμα (στήλης) παντού. Δηλαδή, το <code
class="varname">y0</code> μπορεί να είναι ένα διάνυσμα, οπότε το <code class="varname">f</code> πρ�
�πει να πάρει έναν αριθμό <code class="varname">x</code> και ένα διάνυσμα του ίδιου μεγέθους για το δεύτερο
όρισμα και πρέπει να επιστρέψει ένα διάνυσμα του ίδιου μεγέθους.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
@@ -109,5 +116,5 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Προηγ</a> </td><td width="20%" align="center"><a accesskey="u"
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href="ch11s14.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Συναρτήσεις </td><td
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diff --git a/help/el/html/ch11s14.html b/help/el/html/ch11s14.html
index 92fd75f..4069cee 100644
--- a/help/el/html/ch11s14.html
+++ b/help/el/html/ch11s14.html
@@ -1 +1,26 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Στατιστική</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html"
title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της GEL"><link rel="prev" href="ch11s13.html" title="Επίλυση
εξίσωσης"><link rel="next" href="ch11s15.html" title="Πολυώνυμα"></head><body bgcolor="white" text="black"
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header"><tr><th colspan="3" align="center">Στατιστική</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch11
s15.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Στατιστική</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Average"></a>Average</span></dt><dd><pre
class="synopsis">Average (m)</pre><p>Παραλλαγές: <code class="function">average</code><code
class="function">Mean</code><code class="function">mean</code></p><p>Υπολογίζει τον μέσο όρο ενός ολόκληρου
πίνακα.</p><p>Δείτε <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Ολοκλήρ�
�μα της GaussFunction από 0 μέχρι <code class="varname">x</code> (περιοχή κάτω από την κανονική
καμπύλη).</p><p>Δείτε <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Η συνάρτηση κανονικοποιημένης κατανομής Γκάους (η κανονική καμπύλη).</p><p>Δείτε <a
class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> για
περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-Median"></a>Median</span></dt><dd><pre class="synopsis">Median (m)</pre><p>Παραλλαγές:
<code class="function">median</code></p><p>Υπο�
�ογίζει τον μέσο ενός ολόκληρου πίνακα.</p><p>Δείτε <a class="ulink"
href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> για περισσότερες
πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">stdevp</code></p><p>Υπολογίζει την τυπική απόκλιση πληθυσμού ενός ολόκληρου
πίνακα.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Παραλλαγές: <code class="function">RowMean</code></p><p>Υπολογίζει
τον μέσο όρο κάθε γραμμής σε έναν πίνακα.</p><p>Δείτε <a class="ulink" href="http://mathworld.wolfra
m.com/ArithmeticMean.html" target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span
class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre class="synopsis">RowMedian
(m)</pre><p>Υπολογίζει τον διάμεσο κάθε γραμμής σε έναν πίνακα και επιστρέφει ένα διάνυσμα στήλης με τους
διάμεσους.</p><p>Δείτε <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">rowstdevp</code></p><p>Υπολογίζει τις τυπικές αποκλίσεις πληθυσμού γραμμών εν�
�ς πίνακα και επιστρέφει ένα κάθετο διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">rowstdev</code></p><p>Υπολογίζει τις τυπικές αποκλίσεις γραμμών ενός πίνακα και επιστρέφει
ένα κάθετο διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">stdev</code></p><p>Υπολογίζει την τυπική απόκλιση ενός ολόκληρου
πίνακα.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s13.html">Προ�
�γ</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Πάνω</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch11s15.html">Επόμενο</a></td></tr><tr><td width="40%" align="left"
valign="top">Επίλυση εξίσωσης </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Αρχή</a></td><td width="40%" align="right" valign="top">
Πολυώνυμα</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Στατιστική</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html"
title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της GEL"><link rel="prev" href="ch11s13.html" title="Επίλυση
εξίσωσης"><link rel="next" href="ch11s15.html" title="Πολυώνυμα"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Στατιστική</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος
συναρτήσεων της GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch11
s15.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Στατιστική</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Average"></a>Average</span></dt><dd><pre
class="synopsis">Average (m)</pre><p>Παραλλαγές: <code class="function">average</code><code
class="function">Mean</code><code class="function">mean</code></p><p>Calculate average (the arithmetic mean)
of an entire matrix.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Ολοκλήρωμα της GaussFunction από 0 μέχρι <code
class="varname">x</code> (περιοχή κάτω από την κανονική καμπύλη).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Η συνάρτηση κανονικοποιημένης κατανομής Γκάους (η κανονική καμπύλη).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Παραλλαγές: <code class="function">median</code></p><p>Υπολογίζει τον
μέσο ενός ολόκληρου πίνακα.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">stdevp</code></p><p>Υπολογίζει την τυπική απόκλιση πληθυσμού ενός ολόκληρου
πίνακα.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Παραλλαγές: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Υπολογίζει τον διάμεσο κάθε γραμμής σε έναν πίνακα και επιστρέφει ένα
διάνυσμα στήλης με τους διάμεσους.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">rowstdevp</code></p><p>Υπολογίζει τις τυπικές αποκλίσεις πληθυσμού γραμμών ενός πίνακα και
επιστρέφει ένα κάθετο διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Παραλλαγές: <code
class="function">rowstdev</code></p><p>Υπολογίζει τις τυπικές αποκλίσεις γραμμών ενός πίνακα και επιστρέφει
ένα κάθετο διάνυσμα.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre class="synops
is">StandardDeviation (m)</pre><p>Παραλλαγές: <code class="function">stdev</code></p><p>Υπολογίζει την
τυπική απόκλιση ενός ολόκληρου πίνακα.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Προηγ</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Επίλυση εξίσωσης
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td width="40%"
align="right" valign="top"> Πολυώνυμα</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s15.html b/help/el/html/ch11s15.html
index ec158cf..3c58e8f 100644
--- a/help/el/html/ch11s15.html
+++ b/help/el/html/ch11s15.html
@@ -2,4 +2,7 @@
See
<a class="ulink" href="http://planetmath.org/PolynomialLongDivision"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-IsPoly"></a>IsPoly</span></dt><dd><pre
class="synopsis">IsPoly (p)</pre><p>Ελέγχει αν ένα διάνυσμα μπορεί να χρησιμοποιηθεί ως
πολυώνυμο.</p></dd><dt><span class="term"><a
name="gel-function-MultiplyPoly"></a>MultiplyPoly</span></dt><dd><pre class="synopsis">MultiplyPoly
(p1,p2)</pre><p>Πολλαπλασιάζει δύο πολυώνυμα (ως διανύσματα).</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethodPoly"></a>NewtonsMethodPoly</span></dt><dd><pre
class="synopsis">NewtonsMethodPoly (poly,guess,epsilon,maxn)</pre><p>Βρίσκει μια ρίζα ενός πολυωνύμου
χρησιμοποιώντας τη μέθοδο Νεύτωνα. Το <code class="varname">poly</code> είναι ένα πολυώνυμο ως διάνυσμα και
<code class="varname">guess</code> είναι η αρχική πρόβλεψη. Η συνάρ
τηση επιστρέφει μετά από δύο διαδοχικές τιμές που είναι μέσα στο <code class="varname">epsilon</code> μεταξύ
τους, ή μετά από <code class="varname">maxn</code> προσπάθειες, οπότε η συνάρτηση επιστρέφει <code
class="constant">null</code> που δείχνει αποτυχία.</p><p>Δείτε επίσης <a class="link"
href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a>.</p><p>Παράδειγμα εύρεσης της τετραγωνικής ρίζας του 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethodPoly([-10,0,1],3,10^-10,100)</code></strong>
-</pre><p>Δείτε <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method";
target="_top">Wikipedia</a> για περισσότερες πληροφορίες.</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Δίνει τη δεύτερη πολυωνυμική παράγωγο (ως
διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Δίνει την πολυωνυμική παράγωγο (ως διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Δημιουργεί συνάρτηση από ένα πολυώνυμο (ως διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>Poly
ToString</span></dt><dd><pre class="synopsis">PolyToString (p,var...)</pre><p>Δημιουργεί συμβολοσειρά από
ένα πολυώνυμο (ως διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Αφαιρεί δύο πολυώνυμα (ως διανύσματα).</p></dd><dt><span class="term"><a
name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly
(p)</pre><p>Περικόπτει μηδενικά από ένα πολυώνυμο (ως διάνυσμα).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Προηγ</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s16.html">Επόμενο<
/a></td></tr><tr><td width="40%" align="left" valign="top">Στατιστική </td><td width="20%" align="center"><a
accesskey="h" href="index.html">Αρχή</a></td><td width="40%" align="right" valign="top"> Θεωρία
συνόλων</td></tr></table></div></body></html>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Δίνει τη δεύτερη πολυωνυμική παράγωγο (ως
διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Δίνει την πολυωνυμική παράγωγο (ως διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Δημιουργεί συνάρτηση από ένα πολυώνυμο (ως διάνυσμα).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Δημιουργεί συμβολοσειρά από ένα πολυώνυμο (ως δι�
�νυσμα).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Αφαιρεί δύο πολυώνυμα (ως διανύσματα).</p></dd><dt><span class="term"><a
name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly
(p)</pre><p>Περικόπτει μηδενικά από ένα πολυώνυμο (ως διάνυσμα).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Προηγ</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Στατιστική </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td
<td width="40%" align="right" valign="top"> Θεωρία συνόλων</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s18.html b/help/el/html/ch11s18.html
index 8fdbe57..460c8ea 100644
--- a/help/el/html/ch11s18.html
+++ b/help/el/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Διάφορα</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html" title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της
GEL"><link rel="prev" href="ch11s17.html" title="Αντιμεταθετική άλγεβρα"><link rel="next" href="ch11s19.html"
title="Συμβολικές πράξεις"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Διάφορα</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος συναρτήσεων της
GEL</th><td width="20%" align="right"> <a accessk
ey="n" href="ch11s19.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Διάφορα</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Μετατρέπει ένα διάνυσμα τιμών ASCII σε συμβολοσειρά.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Μετατρέπει ένα διάνυσμα τιμών αλφαβήτου με βάση το 0
(θέσεις στη συμβολοσειρά αλφαβήτου) σε συμβολοσειρά.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</
span></dt><dd><pre class="synopsis">StringToASCII (str)</pre><p>Μετατρέπει μια συμβολοσειρά σε διάνυσμα
τιμών ASCII.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Μετατρέπει μια συμβολοσειρά σε διάνυσμα τιμών
αλφαβήτου με βάση το 0 (θέσεις στη συμβολοσειρά αλφαβήτου), -1 για άγνωστα
γράμματα.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Αντιμ�
�ταθετική άλγεβρα </td><td width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td
width="40%" align="right" valign="top"> Συμβολικές πράξεις</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Διάφορα</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Εγχειρίδιο Genius"><link rel="up" href="ch11.html" title="Κεφάλαιο 11. Κατάλογος συναρτήσεων της
GEL"><link rel="prev" href="ch11s17.html" title="Αντιμεταθετική άλγεβρα"><link rel="next" href="ch11s19.html"
title="Συμβολικές πράξεις"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Διάφορα</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Προηγ</a> </td><th width="60%" align="center">Κεφάλαιο 11. Κατάλογος συναρτήσεων της
GEL</th><td width="20%" align="right"> <a accessk
ey="n" href="ch11s19.html">Επόμενο</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Διάφορα</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Προηγ</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Πάνω</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Επόμενο</a></td></tr><tr><td width="40%" align="left" valign="top">Αντιμεταθετική άλγεβρα
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Αρχή</a></td><td width="40%"
align="right" valign="top"> Συμβολικές πράξεις</td></tr></table></div></body></html>
diff --git a/help/el/html/ch11s20.html b/help/el/html/ch11s20.html
index 8f9c998..68b7315 100644
--- a/help/el/html/ch11s20.html
+++ b/help/el/html/ch11s20.html
@@ -38,7 +38,7 @@
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
</pre><p>
@@ -80,7 +80,7 @@
Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
</pre><p>
@@ -166,7 +166,7 @@
<code class="varname">n</code> by 3 matrix for a longer polyline.
</p><p>
Extra parameters can be added to specify line color, thickness,
- arrows, the plotting window, or legend.
+ the plotting window, or legend.
You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
<strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>,
diff --git a/help/el/html/index.html b/help/el/html/index.html
index 0753826..30f92f3 100644
--- a/help/el/html/index.html
+++ b/help/el/html/index.html
@@ -1,5 +1,5 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Εγχειρίδιο
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Εγχειρίδιο για το εργαλείο μαθηματικών."><link rel="home" href="index.html" title="Εγχειρίδιο
Genius"><link rel="next" href="ch01.html" title="Κεφάλαιο 1. Εισαγωγή"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Εγχειρίδιο Genius</th></tr><tr><td width="20%"
align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Επόμενο</a></td></tr></table><hr></div><div lang="el" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Εγχειρίδιο Ge
nius</h1></div><div><div class="authorgroup"><div class="author"><h3 class="author"><span
class="firstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Πανεπιστήμιο πολιτείας Οκλαχόμα<br></span><div class="address"><p> <code class="email"><<a
class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code> </p></div></div></div><div
class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">Πανεπιστήμιο του
Κουινσλάντ, Αυστραλία<br></span><div class="address"><p> <code class="email"><<a class="email"
href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">This manual describes version 1.0.22 of Genius.
- </p></div><div><p class="copyright">Πνευματικά Δικαιώματα © 1997-2016 Jiří (George)
Lebl</p></div><div><p class="copyright">Πνευματικά Δικαιώματα © 2004 Kai Willadsen</p></div><div><p
class="copyright">Πνευματικά Δικαιώματα © 2013 Δημήτρης Σπίγγος (dmtrs32 gmail com)</p></div><div><p
class="copyright">Πνευματικά Δικαιώματα © 2014 Μαρία Μαυρίδου (mavridou gmail com)</p></div><div><div
class="legalnotice"><a name="legalnotice"></a><p>Χορηγείται άδεια αντιγραφής, διανομής και/ή τροποποίησης του
παρόντος εγγράφου υπό τους όρους της έκδοσης 1.1 της Ελεύθερης Άδειας Τεκμηρίωσης GNU (GFDL), ή οποιασδήποτε
μεταγενέστερης έκδοσής αυτής από το Ίδρυμα Ελεύθερου Λογισμικού (FSF), χωρ
ίς αμετάβλητες ενότητες, κείμενα εμπροσθοφύλλου και κείμενα οπισθοφύλλου. Αντίγραφο της άδειας GFDL είναι
διαθέσιμο στον ακόλουθο <a class="ulink" href="ghelp:fdl" target="_top">σύνδεσμο</a>, ή στο αρχείο
COPYING-DOCS που διανέμεται μαζί με το παρόν εγχειρίδιο.</p><p>Αυτό το εγχειρίδιο αποτελεί μέρος της συλλογής
εγχειριδίων του GNOME που διανέμονται υπό τους όρους της GFDL. Αν επιθυμείτε να διανείμετε το παρόν
εγχειρίδιο ξεχωριστά από τη συλλογή, οφείλετε να προσθέσετε στο εγχειρίδιο αντίγραφο της άδειας χρήσης, όπως
προβλέπεται στην ενότητα 6 της άδειας.</p><p>Πολλές από �
�ις ονομασίες που χρησιμοποιούνται από εταιρείες για την διαφοροποίηση των προϊόντων και υπηρεσιών τους
έχουν καταχωρηθεί ως εμπορικά σήματα. Σε όποιο σημείο της τεκμηρίωσης GNOME τυχόν εμφανίζονται αυτές οι
ονομασίες, και εφόσον τα μέλη του Έργου τεκμηρίωσης GNOME έχουν λάβει γνώση αυτών των εμπορικών σημάτων, οι
ονομασίες ή τα αρχικά αυτών θα γράφονται με κεφαλαίους χαρακτήρες.</p><p>ΤΟ ΠΑΡΟΝ ΕΓΓΡΑΦΟ ΚΑΙ ΟΙ ΤΡΟΠΟΙΗΜΕΝΕΣ
ΕΚΔΟΣΕΙΣ ΑΥΤΟΥ ΠΑΡΕΧΟΝΤΑΙ ΥΠΟ ΤΟΥΣ ΟΡΟΥΣ ΤΗΣ ΕΛΕΥΘΕΡΗΣ ΑΔΕΙΑΣ ΤΕΚΜΗΡΙΩΣΗΣ GNU (GFDL) ΚΑΙ ΜΕ ΤΗΝ ΠΕΡΑΙΤΕΡΩ
ΔΙΕΥΚΡΙΝΙΣΗ ΟΤΙ: </
p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>ΤΟ ΠΑΡΟΝ ΕΓΓΡΑΦΟ
ΠΑΡΕΧΕΤΑΙ "ΩΣ ΕΧΕΙ", ΧΩΡΙΣ ΟΠΟΙΑΔΗΠΟΤΕ ΑΛΛΗ ΕΓΓΥΗΣΗ, ΕΙΤΕ ΡΗΤΗ ΕΙΤΕ ΣΙΩΠΗΡΗ, ΣΥΜΠΕΡΙΛΑΜΒΑΝΟΜΕΝΗΣ, ΧΩΡΙΣ
ΠΕΡΙΟΡΙΣΜΟ, ΤΗΣ ΕΓΓΥΗΣΗΣ ΟΤΙ ΤΟ ΕΓΓΡΑΦΟ, Ή Η ΤΡΟΠΟΠΟΙΗΜΕΝΗ ΕΚΔΟΣΗ ΑΥΤΟΥ, ΕΙΝΑΙ ΕΜΠΟΡΕΥΣΙΜΟ, ΚΑΤΑΛΛΗΛΟ ΓΙΑ
ΕΙΔΙΚΟ ΣΚΟΠΟ ΚΑΙ ΔΕΝ ΠΡΟΣΒΑΛΛΕΙ ΔΙΚΑΙΩΜΑΤΑ ΤΡΙΤΩΝ. Ο ΧΡΗΣΤΗΣ ΑΝΑΛΑΜΒΑΝΕΙ ΕΞ ΟΛΟΚΛΗΡΟΥ ΤΗΝ ΕΘΥΝΗ ΩΣ ΠΡΟΣ ΤΗΝ
ΠΟΙΟΤΗΤΑ, ΤΗΝ ΑΚΡΙΒΕΙΑ ΚΑΙ ΤΗΝ ΧΡΗΣΗ ΤΟΥ ΕΓΓΡΑΦΟΥ Ή ΤΗΣ ΤΡΟΠΟΠΟΙΗΜΕΝΗΣ ΕΚΔΟΣΗΣ ΑΥΤΟΥ. ΣΕ ΠΕΡΙΠΤΩΣΗ ΠΟΥ
ΟΠΟΙΟΔΗΠΟΤΕ ΕΓΓΡΑΦΟ Ή ΤΡΟΠΟΠΟΙΗΜΕΝΗ ΕΚΔΟΣΗ ΑΥΤΟΥ ΑΠΟΔΕΙΧ
ΘΟΥΝ ΕΛΑΤΤΩΜΑΤΙΚΑ ΚΑΘ' ΟΙΟΝΔΗΠΟΤΕ ΤΡΟΠΟ, Ο ΧΡΗΣΤΗΣ (ΚΑΙ ΟΧΙ Ο ΑΡΧΙΚΟΣ ΣΥΓΓΡΑΦΕΑΣ, ΔΗΜΙΟΥΡΓΟΣ Ή ΟΠΟΙΟΣΔΗΠΟΤΕ
ΣΥΝΤΕΛΕΣΤΗΣ) ΑΝΑΛΑΜΒΑΝΕΙ ΤΟ ΚΟΣΤΟΣ ΟΠΟΙΑΣΔΗΠΟΤΕ ΑΝΑΓΚΑΙΑΣ ΣΥΝΤΗΡΗΣΗΣ, ΕΠΙΣΚΕΥΗΣ Ή ΔΙΟΡΘΩΣΗΣ. Η ΠΑΡΟΥΣΑ
ΑΠΟΠΟΙΗΣΗ ΕΓΓΥΗΣΗΣ ΑΠΟΤΕΛΕΙ ΑΝΑΠΟΣΠΑΣΤΟ ΜΕΡΟΣ ΤΗΣ ΑΔΕΙΑΣ. ΔΕΝ ΕΠΙΤΡΕΠΕΤΑΙ ΟΥΔΕΜΙΑ ΧΡΗΣΗ ΤΟΥ ΕΓΓΡΑΦΟΥ Ή
ΤΡΟΠΟΠΟΙΗΜΕΝΩΝ ΕΚΔΟΣΕΩΝ ΑΥΤΟΥ ΣΥΜΦΩΝΑ ΜΕ ΤΟΥΣ ΟΡΟΥΣ ΤΗΣ ΠΑΡΟΥΣΑΣ, ΠΑΡΑ ΜΟΝΟ ΕΑΝ ΣΥΝΟΔΕΥΕΤΑΙ ΑΠΟ ΤΗΝ ΑΠΟΠΟΙΗΣΗ
ΕΓΓΥΗΣΗΣ, ΚΑΙ</p></li><li class="listitem"><p>Ο ΔΗΜΙΟΥΡΓΟΣ, Ο ΑΡΧΙΚΟΣ ΣΥΓΓΡΑΦΕΑΣ, ΟΙ ΣΥΝΤΕΛΕΣΤΕΣ Ή ΟΙ
ΔΙΑΝΟΜΕΙΣ ΤΟΥ ΕΓΓΡΑΦΟΥ Ή Τ�
�ΟΠΟΠΟΙΗΜΕΝΗΣ ΕΚΔΟΣΗΣ ΑΥΤΟΥ, ΚΑΘΩΣ ΚΑΙ ΟΙ ΠΡΟΜΗΘΕΥΤΕΣ ΟΠΟΙΩΝΔΗΠΟΤΕ ΕΚ ΤΩΝ ΠΡΟΑΝΑΦΕΡΟΜΕΝΩΝ ΜΕΡΩΝ, ΔΕΝ
ΕΥΘΥΝΟΝΤΑΙ ΕΝΑΝΤΙ ΟΙΟΥΔΗΠΟΤΕ, ΣΕ ΚΑΜΙΑ ΠΕΡΙΠΤΩΣΗ ΚΑΙ ΥΠΟ ΚΑΜΙΑ ΕΡΜΗΝΕΙΑ ΝΟΜΟΥ, ΕΙΤΕ ΕΞ ΑΔΙΚΟΠΡΑΞΙΑΣ
(ΣΥΜΠΕΡΙΛΑΜΒΑΝΟΜΕΝΗΣ ΤΗΣ ΑΜΕΛΕΙΑΣ) ΕΙΤΕ ΣΤΟ ΠΛΑΙΣΙΟ ΣΥΜΒΑΤΙΚΗΣ Ή ΑΛΛΗΣ ΥΠΟΧΡΕΩΣΗΣ, ΓΙΑ ΤΥΧΟΝ ΑΜΕΣΕΣ, ΕΜΜΕΣΕΣ,
ΕΙΔΙΚΕΣ, ΤΥΧΑΙΕΣ Ή ΣΥΝΕΠΑΚΟΛΟΥΘΕΣ ΖΗΜΙΕΣ ΟΠΟΙΑΣΔΗΠΟΤΕ ΜΟΡΦΗΣ, ΣΥΜΠΕΡΙΛΑΜΒΑΝΟΜΕΝΩΝ, ΧΩΡΙΣ ΠΕΡΙΟΡΙΣΜΟ, ΖΗΜΙΩΝ
ΛΟΓΩ ΑΠΩΛΕΙΑΣ ΦΗΜΗΣ ΚΑΙ ΠΕΛΑΤΕΙΑΣ, ΔΙΑΚΟΠΗΣ ΕΡΓΑΣΙΩΝ, ΔΥΣΛΕΙΤΟΥΡΓΙΑΣ Ή ΒΛΑΒΗΣ ΗΛΕΚΤΡΟΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ, Ή
ΚΑΘΕ ΑΛΛΗΣ �
�ΗΜΙΑΣ Ή ΑΠΩΛΕΙΑΣ ΠΟΥ ΟΦΕΙΛΕΤΑΙ Ή ΣΧΕΤΙΖΕΤΑΙ ΜΕ ΤΗΝ ΧΡΗΣΗ ΤΟΥ ΕΓΓΡΑΦΟΥ ΚΑΙ ΤΩΝ ΤΡΟΠΟΠΟΙΗΜΕΝΩΝ ΕΚΔΟΣΕΩΝ
ΑΥΤΟΥ, ΑΚΟΜΑ ΚΑΙ ΑΝ ΤΑ ΩΣ ΑΝΩ ΜΕΡΗ ΕΙΧΑΝ ΛΑΒΕΙ ΓΝΩΣΗ ΤΗΣ ΠΙΘΑΝΟΤΗΤΑΣ ΠΡΟΚΛΗΣΗΣ ΤΕΤΟΙΩΝ
ΖΗΜΙΩΝ.</p></li></ol></div></div></div><div><div class="legalnotice"><a name="idm45617557667008"></a><p
class="legalnotice-title"><b>Aνάδραση</b></p><p>
+ </p></div><div><p class="copyright">Πνευματικά Δικαιώματα © 1997-2016 Jiří (George)
Lebl</p></div><div><p class="copyright">Πνευματικά Δικαιώματα © 2004 Kai Willadsen</p></div><div><p
class="copyright">Πνευματικά Δικαιώματα © 2013 Δημήτρης Σπίγγος (dmtrs32 gmail com)</p></div><div><p
class="copyright">Πνευματικά Δικαιώματα © 2014 Μαρία Μαυρίδου (mavridou gmail com)</p></div><div><div
class="legalnotice"><a name="legalnotice"></a><p>Χορηγείται άδεια αντιγραφής, διανομής και/ή τροποποίησης του
παρόντος εγγράφου υπό τους όρους της έκδοσης 1.1 της Ελεύθερης Άδειας Τεκμηρίωσης GNU (GFDL), ή οποιασδήποτε
μεταγενέστερης έκδοσής αυτής από το Ίδρυμα Ελεύθερου Λογισμικού (FSF), χωρ
ίς αμετάβλητες ενότητες, κείμενα εμπροσθοφύλλου και κείμενα οπισθοφύλλου. Αντίγραφο της άδειας GFDL είναι
διαθέσιμο στον ακόλουθο <a class="ulink" href="ghelp:fdl" target="_top">σύνδεσμο</a>, ή στο αρχείο
COPYING-DOCS που διανέμεται μαζί με το παρόν εγχειρίδιο.</p><p>Αυτό το εγχειρίδιο αποτελεί μέρος της συλλογής
εγχειριδίων του GNOME που διανέμονται υπό τους όρους της GFDL. Αν επιθυμείτε να διανείμετε το παρόν
εγχειρίδιο ξεχωριστά από τη συλλογή, οφείλετε να προσθέσετε στο εγχειρίδιο αντίγραφο της άδειας χρήσης, όπως
προβλέπεται στην ενότητα 6 της άδειας.</p><p>Πολλές από �
�ις ονομασίες που χρησιμοποιούνται από εταιρείες για την διαφοροποίηση των προϊόντων και υπηρεσιών τους
έχουν καταχωρηθεί ως εμπορικά σήματα. Σε όποιο σημείο της τεκμηρίωσης GNOME τυχόν εμφανίζονται αυτές οι
ονομασίες, και εφόσον τα μέλη του Έργου τεκμηρίωσης GNOME έχουν λάβει γνώση αυτών των εμπορικών σημάτων, οι
ονομασίες ή τα αρχικά αυτών θα γράφονται με κεφαλαίους χαρακτήρες.</p><p>ΤΟ ΠΑΡΟΝ ΕΓΓΡΑΦΟ ΚΑΙ ΟΙ ΤΡΟΠΟΙΗΜΕΝΕΣ
ΕΚΔΟΣΕΙΣ ΑΥΤΟΥ ΠΑΡΕΧΟΝΤΑΙ ΥΠΟ ΤΟΥΣ ΟΡΟΥΣ ΤΗΣ ΕΛΕΥΘΕΡΗΣ ΑΔΕΙΑΣ ΤΕΚΜΗΡΙΩΣΗΣ GNU (GFDL) ΚΑΙ ΜΕ ΤΗΝ ΠΕΡΑΙΤΕΡΩ
ΔΙΕΥΚΡΙΝΙΣΗ ΟΤΙ: </
p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>ΤΟ ΠΑΡΟΝ ΕΓΓΡΑΦΟ
ΠΑΡΕΧΕΤΑΙ "ΩΣ ΕΧΕΙ", ΧΩΡΙΣ ΟΠΟΙΑΔΗΠΟΤΕ ΑΛΛΗ ΕΓΓΥΗΣΗ, ΕΙΤΕ ΡΗΤΗ ΕΙΤΕ ΣΙΩΠΗΡΗ, ΣΥΜΠΕΡΙΛΑΜΒΑΝΟΜΕΝΗΣ, ΧΩΡΙΣ
ΠΕΡΙΟΡΙΣΜΟ, ΤΗΣ ΕΓΓΥΗΣΗΣ ΟΤΙ ΤΟ ΕΓΓΡΑΦΟ, Ή Η ΤΡΟΠΟΠΟΙΗΜΕΝΗ ΕΚΔΟΣΗ ΑΥΤΟΥ, ΕΙΝΑΙ ΕΜΠΟΡΕΥΣΙΜΟ, ΚΑΤΑΛΛΗΛΟ ΓΙΑ
ΕΙΔΙΚΟ ΣΚΟΠΟ ΚΑΙ ΔΕΝ ΠΡΟΣΒΑΛΛΕΙ ΔΙΚΑΙΩΜΑΤΑ ΤΡΙΤΩΝ. Ο ΧΡΗΣΤΗΣ ΑΝΑΛΑΜΒΑΝΕΙ ΕΞ ΟΛΟΚΛΗΡΟΥ ΤΗΝ ΕΘΥΝΗ ΩΣ ΠΡΟΣ ΤΗΝ
ΠΟΙΟΤΗΤΑ, ΤΗΝ ΑΚΡΙΒΕΙΑ ΚΑΙ ΤΗΝ ΧΡΗΣΗ ΤΟΥ ΕΓΓΡΑΦΟΥ Ή ΤΗΣ ΤΡΟΠΟΠΟΙΗΜΕΝΗΣ ΕΚΔΟΣΗΣ ΑΥΤΟΥ. ΣΕ ΠΕΡΙΠΤΩΣΗ ΠΟΥ
ΟΠΟΙΟΔΗΠΟΤΕ ΕΓΓΡΑΦΟ Ή ΤΡΟΠΟΠΟΙΗΜΕΝΗ ΕΚΔΟΣΗ ΑΥΤΟΥ ΑΠΟΔΕΙΧ
ΘΟΥΝ ΕΛΑΤΤΩΜΑΤΙΚΑ ΚΑΘ' ΟΙΟΝΔΗΠΟΤΕ ΤΡΟΠΟ, Ο ΧΡΗΣΤΗΣ (ΚΑΙ ΟΧΙ Ο ΑΡΧΙΚΟΣ ΣΥΓΓΡΑΦΕΑΣ, ΔΗΜΙΟΥΡΓΟΣ Ή ΟΠΟΙΟΣΔΗΠΟΤΕ
ΣΥΝΤΕΛΕΣΤΗΣ) ΑΝΑΛΑΜΒΑΝΕΙ ΤΟ ΚΟΣΤΟΣ ΟΠΟΙΑΣΔΗΠΟΤΕ ΑΝΑΓΚΑΙΑΣ ΣΥΝΤΗΡΗΣΗΣ, ΕΠΙΣΚΕΥΗΣ Ή ΔΙΟΡΘΩΣΗΣ. Η ΠΑΡΟΥΣΑ
ΑΠΟΠΟΙΗΣΗ ΕΓΓΥΗΣΗΣ ΑΠΟΤΕΛΕΙ ΑΝΑΠΟΣΠΑΣΤΟ ΜΕΡΟΣ ΤΗΣ ΑΔΕΙΑΣ. ΔΕΝ ΕΠΙΤΡΕΠΕΤΑΙ ΟΥΔΕΜΙΑ ΧΡΗΣΗ ΤΟΥ ΕΓΓΡΑΦΟΥ Ή
ΤΡΟΠΟΠΟΙΗΜΕΝΩΝ ΕΚΔΟΣΕΩΝ ΑΥΤΟΥ ΣΥΜΦΩΝΑ ΜΕ ΤΟΥΣ ΟΡΟΥΣ ΤΗΣ ΠΑΡΟΥΣΑΣ, ΠΑΡΑ ΜΟΝΟ ΕΑΝ ΣΥΝΟΔΕΥΕΤΑΙ ΑΠΟ ΤΗΝ ΑΠΟΠΟΙΗΣΗ
ΕΓΓΥΗΣΗΣ, ΚΑΙ</p></li><li class="listitem"><p>Ο ΔΗΜΙΟΥΡΓΟΣ, Ο ΑΡΧΙΚΟΣ ΣΥΓΓΡΑΦΕΑΣ, ΟΙ ΣΥΝΤΕΛΕΣΤΕΣ Ή ΟΙ
ΔΙΑΝΟΜΕΙΣ ΤΟΥ ΕΓΓΡΑΦΟΥ Ή Τ�
�ΟΠΟΠΟΙΗΜΕΝΗΣ ΕΚΔΟΣΗΣ ΑΥΤΟΥ, ΚΑΘΩΣ ΚΑΙ ΟΙ ΠΡΟΜΗΘΕΥΤΕΣ ΟΠΟΙΩΝΔΗΠΟΤΕ ΕΚ ΤΩΝ ΠΡΟΑΝΑΦΕΡΟΜΕΝΩΝ ΜΕΡΩΝ, ΔΕΝ
ΕΥΘΥΝΟΝΤΑΙ ΕΝΑΝΤΙ ΟΙΟΥΔΗΠΟΤΕ, ΣΕ ΚΑΜΙΑ ΠΕΡΙΠΤΩΣΗ ΚΑΙ ΥΠΟ ΚΑΜΙΑ ΕΡΜΗΝΕΙΑ ΝΟΜΟΥ, ΕΙΤΕ ΕΞ ΑΔΙΚΟΠΡΑΞΙΑΣ
(ΣΥΜΠΕΡΙΛΑΜΒΑΝΟΜΕΝΗΣ ΤΗΣ ΑΜΕΛΕΙΑΣ) ΕΙΤΕ ΣΤΟ ΠΛΑΙΣΙΟ ΣΥΜΒΑΤΙΚΗΣ Ή ΑΛΛΗΣ ΥΠΟΧΡΕΩΣΗΣ, ΓΙΑ ΤΥΧΟΝ ΑΜΕΣΕΣ, ΕΜΜΕΣΕΣ,
ΕΙΔΙΚΕΣ, ΤΥΧΑΙΕΣ Ή ΣΥΝΕΠΑΚΟΛΟΥΘΕΣ ΖΗΜΙΕΣ ΟΠΟΙΑΣΔΗΠΟΤΕ ΜΟΡΦΗΣ, ΣΥΜΠΕΡΙΛΑΜΒΑΝΟΜΕΝΩΝ, ΧΩΡΙΣ ΠΕΡΙΟΡΙΣΜΟ, ΖΗΜΙΩΝ
ΛΟΓΩ ΑΠΩΛΕΙΑΣ ΦΗΜΗΣ ΚΑΙ ΠΕΛΑΤΕΙΑΣ, ΔΙΑΚΟΠΗΣ ΕΡΓΑΣΙΩΝ, ΔΥΣΛΕΙΤΟΥΡΓΙΑΣ Ή ΒΛΑΒΗΣ ΗΛΕΚΤΡΟΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ, Ή
ΚΑΘΕ ΑΛΛΗΣ �
�ΗΜΙΑΣ Ή ΑΠΩΛΕΙΑΣ ΠΟΥ ΟΦΕΙΛΕΤΑΙ Ή ΣΧΕΤΙΖΕΤΑΙ ΜΕ ΤΗΝ ΧΡΗΣΗ ΤΟΥ ΕΓΓΡΑΦΟΥ ΚΑΙ ΤΩΝ ΤΡΟΠΟΠΟΙΗΜΕΝΩΝ ΕΚΔΟΣΕΩΝ
ΑΥΤΟΥ, ΑΚΟΜΑ ΚΑΙ ΑΝ ΤΑ ΩΣ ΑΝΩ ΜΕΡΗ ΕΙΧΑΝ ΛΑΒΕΙ ΓΝΩΣΗ ΤΗΣ ΠΙΘΑΝΟΤΗΤΑΣ ΠΡΟΚΛΗΣΗΣ ΤΕΤΟΙΩΝ
ΖΗΜΙΩΝ.</p></li></ol></div></div></div><div><div class="legalnotice"><a name="idm54"></a><p
class="legalnotice-title"><b>Aνάδραση</b></p><p>
To report a bug or make a suggestion regarding the <span class="application">Genius Mathematics
Tool</span>
application or this manual, please visit the
<a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius
diff --git a/help/es/html/ch05s07.html b/help/es/html/ch05s07.html
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@@ -1,6 +1,53 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Lista de operadores
GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch05.html" title="Capítulo 5. Conceptos de
GEL"><link rel="prev" href="ch05s06.html" title="Evaluación modular"><link rel="next" href="ch06.html"
title="Capítulo 6. Programar con GEL"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Lista de operadores GEL</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch05s06.html">Anterior</a> </td><th width="60%" align="center">Capítulo
5. Conceptos de GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><di
v><h2 class="title" style="clear: both"><a name="genius-gel-operator-list"></a>Lista de operadores
GEL</h2></div></div></div><p>Todo en GEL es en realidad una expresión. Las expresiones se encadenan unas tras
otras mediante diferentes operadores. Como hemos visto, incluso el separador es un operador binario en GEL. A
continuación se muestra una lista de los operadores en GEL.</p><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>El separador evalúa <code
class="varname">a</code> y <code class="varname">b</code>, pero sólo devuelve el valor de <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>El operador asignación. Asigna <code
class="varname">b</code> a <code class="varname">a</code> (<code class="varname">a</code> debe ser un <a
class="link" href="ch06s09.html" title="Lvalues">lvalue</a> válido
) (tenga en cuenta que este operador puede equivaler a <code class="literal">==</code> si se usa cuando se
espera una expresión booleana)</p></dd><dt><span class="term"><strong
class="userinput"><code>a:=b</code></strong></span></dt><dd><p>El operador asignación. Asigna <code
class="varname">b</code> a <code class="varname">a</code> (<code class="varname">a</code> debe ser un <a
class="link" href="ch06s09.html" title="Lvalues">lvalue</a> válido). Se diferencia de <code
class="literal">=</code> en que nunca equivale a <code class="literal">==</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>|a|</code></strong></span></dt><dd><p>Valor absoluto. En el caso
de que la expresión sea un número complejo el resultado será su módulo (distancia desde el origen). Por
ejemplo: <strong class="userinput"><code>|3 * e^(1i*pi)|</code></strong> devuelve 3.</p><p>Consulte <a
class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworl
d</a> para obtener más información.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Exponenciación, eleva <code
class="varname">a</code> a la <code class="varname">b</code>-ésima potencia.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Potencia elemento a
elemento. Eleva cada elemento de una matriz <code class="varname">a</code> a la <code
class="varname">b</code>-ésima potencia. O si <code class="varname">b</code> es una matriz del mismo tamaño
que <code class="varname">a</code>, entonces realiza la operación elemento a elemento. Si <code
class="varname">a</code> es un número y <code class="varname">b</code> es una matriz entonces crea una matriz
del mismo tamaño que <code class="varname">b</code> formada por <code class="varname">a</code> elevado a
todas las diferentes potencias de <code class="varname">b</code>.</p></dd><dt><span class="term"><strong
class=
"userinput"><code>a+b</code></strong></span></dt><dd><p>Adición. Suma dos números, matrices, funciones o
cadenas. Si suma una cadena a cualquier valor el resultado es una cadena. Si uno de ellos es una matriz
cuadrada y el otro un número, el número se multiplica por la identidad de la matriz.</p></dd><dt><span
class="term"><strong class="userinput"><code>a-b</code></strong></span></dt><dd><p>Sustracción. Resta dos
números, matrices o funciones.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Multiplicación. Es la multiplicación normal de
matrices.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Multiplicación elemento a elemento si <code
class="varname">a</code> y <code class="varname">b</code> son matrices.</p></dd><dt><span
class="term"><strong class="userinput"><code>a/b</code></strong></span></dt><dd><p>División. Cuando <code
class="varname">a</code> y <co
de class="varname">b</code> son sólo números es la división normal. Cuando son matrices, esto es el
equivalente a <strong class="userinput"><code>a*b^-1</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>División elemento por elemento. Igual que
<strong class="userinput"><code>a/b</code></strong> para números, pero opera elemento por elemento en
matrices.</p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>División hacia atrás. Es lo mismo que <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>División hacia atrás elemento por
elemento.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>El operador mod. No activa el <a class="link"
href="ch05s06.html" title="Evaluación modular">modo modular</a> si
no que simplemente devuelve el resto de <strong
class="userinput"><code>a/b</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>Operador mod elemento por elemento. Devuelve
el resto de <strong class="userinput"><code>a./b</code></strong> elemento por elemento.</p></dd><dt><span
class="term"><strong class="userinput"><code>a mod b</code></strong></span></dt><dd><p>Operación de
evaluación modular. La expresión <code class="varname">a</code> se evalúa módulo <code
class="varname">b</code>. Consulte la <a class="xref" href="ch05s06.html" title="Evaluación
modular">“Evaluación modular”</a>. Algunas de las funciones y operadores se comportan de un modo distinto
cuando trabajan en módulo entero.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Operador factorial. Esto es <strong
class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><sp
an class="term"><strong class="userinput"><code>a!!</code></strong></span></dt><dd><p>Operador doble
factorial. Esto es <strong class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a==b</code></strong></span></dt><dd><p>Operador de igualdad,
devuelve <code class="constant">true</code> o <code class="constant">false</code> dependiendo de si <code
class="varname">a</code> y <code class="varname">b</code> son iguales o no.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Operador de desigualdad,
devuelve <code class="constant">true</code> si <code class="varname">a</code> no es igual a <code
class="varname">b</code>; si lo es, devuelve <code class="constant">false</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Operador distinto
alternativo devuelve <code class="constant">true</code> si <c
ode class="varname">a</code> no es igual a <code class="varname">b</code> en caso contrario devuelve <code
class="constant">false</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Operador menor o igual, devuelve <code
class="constant">true</code> si <code class="varname">a</code> es menor o igual que <code
class="varname">b</code>, si no, devuelve <code class="constant">false</code>. Esto se puede concatenar como
<strong class="userinput"><code>a <= b <= c</code></strong> (también se puede combinar con el operador
menor que).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>Operador mayor o igual, devuelve <code
class="constant">true</code> si <code class="varname">a</code> es mayor o igual que <code
class="varname">b</code>, si no, devuelve <code class="constant">false</code>. Esto se puede concatenar como
<strong class="userinput"><code>a >= b &
gt;= c</code></strong> (también se puede combinar con el operador mayor que).</p></dd><dt><span
class="term"><strong class="userinput"><code>a<b</code></strong></span></dt><dd><p>Operador menor que,
devuelve <code class="constant">true</code> si <code class="varname">a</code> es menor o igual que <code
class="varname">b</code>, si no, devuelve <code class="constant">false</code>. Esto se puede concatenar como
<strong class="userinput"><code>a < b < c</code></strong> (también se puede combinar con el operador
menor o igual que).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>Operador mayor que, devuelve <code
class="constant">true</code> si <code class="varname">a</code> es mayor o igual que <code
class="varname">b</code>, si no, devuelve <code class="constant">false</code>. Esto se puede concatenar como
<strong class="userinput"><code>a > b > c</code></strong> (también se puede combinar con el operado
r mayor o igual que).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Operador de comparación. Si <code
class="varname">a</code> es igual a <code class="varname">b</code> devuelve 0, si <code
class="varname">a</code> es menor que <code class="varname">b</code> devuelve -1 y si <code
class="varname">a</code> es mayor que <code class="varname">b</code> devuelve 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a and b</code></strong></span></dt><dd><p>AND lógico. Devuelve
cierto si <code class="varname">a</code> y <code class="varname">b</code> son ciertos; si no, devuelve falso.
Si se dan números, los números distintos de cero se consideran como «verdadero».</p></dd><dt><span
class="term"><strong class="userinput"><code>a or b</code></strong></span></dt><dd><p>OR lógico. Devuelve
verdadero si <code class="varname">a</code> o <code class="varname">b</code> son verdaderos; si no, devuelve
falso.
Si se dan números, los números distintos de cero se consideran como verdadero.</p></dd><dt><span
class="term"><strong class="userinput"><code>a xor b</code></strong></span></dt><dd><p>X-OR lógico. Devuelve
cierto si <code class="varname">a</code> o <code class="varname">b</code> son ciertos; si no, devuelve falso.
Si se dan números, los números distintos de cero se consideran como «verdadero».</p></dd><dt><span
class="term"><strong class="userinput"><code>not a</code></strong></span></dt><dd><p>NOT lódico. Devuelve la
negación lógica de <code class="varname">a</code></p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>Operador de negación. Devuelve el negativo de un
número o una matriz (en una matriz, funciona de acuerdo al elemento).</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Referencia de variables (pasar una
referencia a una variable). Consulte <a clas
s="xref" href="ch06s08.html" title="Referencias">“Referencias”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>Desreferenciar una variable (para acceder a una
variable referenciada). Consulte la <a class="xref" href="ch06s08.html"
title="Referencias">“Referencias”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Transpuesta conjugada de una matriz. Significa
que las filas y columnas se intercambian y se toman la conjugada compleja de todas las entradas. Esto es, si
el elemento i,j de <code class="varname">a</code> es x+iy, entonces el elemento j,i de <strong
class="userinput"><code>a'</code></strong> es x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Transpuesta de matriz, no conjuga las entradas.
Esto significa, el elemento i,j de <code class="varname">a</code> se convierte en el elemento j,i de <
strong class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>Devuelve el elemento en la fila <code
class="varname">b</code> y columna <code class="varname">c</code>. Si <code class="varname">b</code>, <code
class="varname">c</code> son vectores, devuelve las correspondientes filas, columnas o
submatrices.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Devuelve la fila de la matriz (o múltiples
filas si <code class="varname">b</code> es un vector).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Igual que el anterior</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Devuelve la columna de
la matriz (o columnas si <code class="varname">c</code> es un vector).</p></dd><dt><span class="term"><strong
class="userinput"><
code>a@(:,c)</code></strong></span></dt><dd><p>Igual que el anterior</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Obtiene un elemento de una matriz tratándola
como vector. Recorre la matriz por filas.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Crea un vector con valores de <code
class="varname">a</code> a <code class="varname">b</code> (o específica una región de filas o columnas para
el operador <code class="literal">@</code>). Por ejemplo para obtener las filas 2 a 4 de la matriz <code
class="varname">A</code> se podría hacer </p><pre class="programlisting">A@(2:4,)
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Lista de operadores
GEL</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch05.html" title="Capítulo 5. Conceptos de
GEL"><link rel="prev" href="ch05s06.html" title="Evaluación modular"><link rel="next" href="ch06.html"
title="Capítulo 6. Programar con GEL"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Lista de operadores GEL</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch05s06.html">Anterior</a> </td><th width="60%" align="center">Capítulo
5. Conceptos de GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><di
v><h2 class="title" style="clear: both"><a name="genius-gel-operator-list"></a>Lista de operadores
GEL</h2></div></div></div><p>Todo en GEL es en realidad una expresión. Las expresiones se encadenan unas tras
otras mediante diferentes operadores. Como hemos visto, incluso el separador es un operador binario en GEL. A
continuación se muestra una lista de los operadores en GEL.</p><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>El separador evalúa <code
class="varname">a</code> y <code class="varname">b</code>, pero sólo devuelve el valor de <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>El operador asignación. Asigna <code
class="varname">b</code> a <code class="varname">a</code> (<code class="varname">a</code> debe ser un <a
class="link" href="ch06s09.html" title="Lvalues">lvalue</a> válido
) (tenga en cuenta que este operador puede equivaler a <code class="literal">==</code> si se usa cuando se
espera una expresión booleana)</p></dd><dt><span class="term"><strong
class="userinput"><code>a:=b</code></strong></span></dt><dd><p>El operador asignación. Asigna <code
class="varname">b</code> a <code class="varname">a</code> (<code class="varname">a</code> debe ser un <a
class="link" href="ch06s09.html" title="Lvalues">lvalue</a> válido). Se diferencia de <code
class="literal">=</code> en que nunca equivale a <code class="literal">==</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>|a|</code></strong></span></dt><dd><p>Valor absoluto. En el caso
de que la expresión sea un número complejo el resultado será su módulo (distancia desde el origen). Por
ejemplo: <strong class="userinput"><code>|3 * e^(1i*pi)|</code></strong> devuelve 3.</p><p>Consulte <a
class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworl
d</a> para obtener más información.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Exponenciación, eleva <code
class="varname">a</code> a la <code class="varname">b</code>-ésima potencia.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Potencia elemento a
elemento. Eleva cada elemento de una matriz <code class="varname">a</code> a la <code
class="varname">b</code>-ésima potencia. O si <code class="varname">b</code> es una matriz del mismo tamaño
que <code class="varname">a</code>, entonces realiza la operación elemento a elemento. Si <code
class="varname">a</code> es un número y <code class="varname">b</code> es una matriz entonces crea una matriz
del mismo tamaño que <code class="varname">b</code> formada por <code class="varname">a</code> elevado a
todas las diferentes potencias de <code class="varname">b</code>.</p></dd><dt><span class="term"><strong
class=
"userinput"><code>a+b</code></strong></span></dt><dd><p>Adición. Suma dos números, matrices, funciones o
cadenas. Si suma una cadena a cualquier valor el resultado es una cadena. Si uno de ellos es una matriz
cuadrada y el otro un número, el número se multiplica por la identidad de la matriz.</p></dd><dt><span
class="term"><strong class="userinput"><code>a-b</code></strong></span></dt><dd><p>Sustracción. Resta dos
números, matrices o funciones.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Multiplicación. Es la multiplicación normal de
matrices.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Multiplicación elemento a elemento si <code
class="varname">a</code> y <code class="varname">b</code> son matrices.</p></dd><dt><span
class="term"><strong class="userinput"><code>a/b</code></strong></span></dt><dd><p>División. Cuando <code
class="varname">a</code> y <co
de class="varname">b</code> son sólo números es la división normal. Cuando son matrices, esto es el
equivalente a <strong class="userinput"><code>a*b^-1</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>División elemento por elemento. Igual que
<strong class="userinput"><code>a/b</code></strong> para números, pero opera elemento por elemento en
matrices.</p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>División hacia atrás. Es lo mismo que <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>División hacia atrás elemento por
elemento.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>
+ The mod operator. This does not turn on the <a class="link" href="ch05s06.html"
title="Evaluación modular">modular mode</a>, but
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>Operación de evaluación modular. La expresión <code
class="varname">a</code> se evalúa módulo <code class="varname">b</code>. Consulte la <a class="xref"
href="ch05s06.html" title="Evaluación modular">“Evaluación modular”</a>. Algunas de las funciones y
operadores se comportan de un modo distinto cuando trabajan en módulo entero.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!</code></strong></span></dt><dd><p>Operador factorial. Esto es
<strong class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Operador doble factorial. Esto es <strong
class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p>Operador de igualdad, dev
uelve <code class="constant">true</code> o <code class="constant">false</code> dependiendo de si <code
class="varname">a</code> y <code class="varname">b</code> son iguales o no.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Operador de desigualdad,
devuelve <code class="constant">true</code> si <code class="varname">a</code> no es igual a <code
class="varname">b</code>; si lo es, devuelve <code class="constant">false</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Operador distinto
alternativo devuelve <code class="constant">true</code> si <code class="varname">a</code> no es igual a <code
class="varname">b</code> en caso contrario devuelve <code class="constant">false</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Operador menor o
igual, devuelve <code class="constant">true</code> si <c
ode class="varname">a</code> es menor o igual que <code class="varname">b</code>, si no, devuelve <code
class="constant">false</code>. Esto se puede concatenar como <strong class="userinput"><code>a <= b <=
c</code></strong> (también se puede combinar con el operador menor que).</p></dd><dt><span
class="term"><strong class="userinput"><code>a>=b</code></strong></span></dt><dd><p>
+ Greater than or equal operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than or equal to
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
+ (and they can also be combined with the greater than operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
+ Less than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ less than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
+ (they can also be combined with the less than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
+ Greater than operator,
+ returns <code class="constant">true</code> if <code class="varname">a</code> is
+ greater than
+ <code class="varname">b</code> else returns <code class="constant">false</code>.
+ These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
+ (they can also be combined with the greater than or equal to operator).
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Operador de comparación. Si <code
class="varname">a</code> es igual a <code class="varname">b</code> devuelve 0, si <code
class="varname">a</code> es menor que <code class="varname">b</code> devuelve -1 y si <code
class="varname">a</code> es mayor que <code class="varname">b</code> devuelve 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a and b</code></strong></span></dt><dd><p>AND lógico. Devuelve
cierto si <code class="varname">a</code> y <code class="varname">b</code> son ciertos; si no, devuelve falso.
Si se dan números, los números distintos de cero se consideran como «verdadero».</p></dd><dt><span
class="term"><strong class="userinput"><code>a or b</code></strong></span></dt><dd><p>OR lógico. Devuelve
verdadero si <code class="varname">a</code> o <code class="varname">b</code> son verdaderos; si no, devuelve
falso. Si se dan
números, los números distintos de cero se consideran como verdadero.</p></dd><dt><span class="term"><strong
class="userinput"><code>a xor b</code></strong></span></dt><dd><p>
+ Logical xor.
+ Returns true if exactly one of
+ <code class="varname">a</code> or <code class="varname">b</code> is true,
+ else returns false. If given numbers, nonzero numbers
+ are treated as true.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>Operador de negación. Devuelve el negativo de un
número o una matriz (en una matriz, funciona de acuerdo al elemento).</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Referencia de variables (pasar una
referencia a una variable). Consulte <a class="xref" href="ch06s08.html"
title="Referencias">“Referencias”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>Desreferenciar una variable (para acceder a una
variable referenciada). Consulte la <a class="xref" href="ch06s08.html"
title="Referencias">“Referencias”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Transpuesta conjugada de una matriz. Significa
que las filas y columnas se intercambian y se toman la conjugada compleja de todas las entr
adas. Esto es, si el elemento i,j de <code class="varname">a</code> es x+iy, entonces el elemento j,i de
<strong class="userinput"><code>a'</code></strong> es x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Transpuesta de matriz, no conjuga las entradas.
Esto significa, el elemento i,j de <code class="varname">a</code> se convierte en el elemento j,i de <strong
class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>
+ Get element of a matrix in row <code class="varname">b</code> and column
+ <code class="varname">c</code>. If <code class="varname">b</code>,
+ <code class="varname">c</code> are vectors, then this gets the corresponding
+ rows, columns or submatrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Devuelve la fila de la matriz (o múltiples
filas si <code class="varname">b</code> es un vector).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Igual que el anterior</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Devuelve la columna de
la matriz (o columnas si <code class="varname">c</code> es un vector).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Igual que el anterior</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Obtiene un elemento de
una matriz tratándola como vector. Recorre la matriz por filas.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Crea un vector con valores de
<code class="varname">a</code> a <code class="varname">b</code> (o específica una región de filas o columnas
para el operador <code class="literal">@</code>). Por ejemplo para obtener las filas 2 a 4 de la matriz <code
class="varname">A</code> se podría hacer </p><pre class="programlisting">A@(2:4,)
</pre><p> ya que <strong class="userinput"><code>2:4</code></strong> devolverá el vector <strong
class="userinput"><code>[2,3,4]</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b:c</code></strong></span></dt><dd><p>Crea un vector con valores desde <code
class="varname">a</code> a <code class="varname">c</code> usando <code class="varname">b</code> como paso.
Por ejemplo: </p><pre class="programlisting">genius> 1:2:9
=
`[1, 3, 5, 7, 9]
-</pre><p>Cuando los números implicados son números en coma flotante, por ejemplo <strong
class="userinput"><code>1.0:0.4:3.0</code></strong>, la salida es lo que se espera a pesar de la adición de
0,4 a 1,0 cinco veces es en realidad sólo un poco más de 3,0 debido a la forma en que los números de coma
flotante se almacenan en la base 2 (no hay 0.4, el número real almacenado es sólo ligeramente más grande). La
forma en que se maneja es el mismo que en los bucles «for», «sum», y «prod». Si el final está dentro de
<strong class="userinput"><code>2^-20</code></strong> veces el tamaño de paso del punto final, se utiliza el
punto final y suponemos que no eran errores de redondeo. Esto no es perfecto, pero maneja la mayoría de los
casos. Esta comprobación se realiza sólo desde la versión 1.0.18 en adelante, así que la ejecución de su
código puede ser diferente en las versiones anteriores. Si quiere evitar este problema, utilice los números
racionales reales
, posiblemente usando el <code class="function">float</code> si quiere obtener los números de punto flotante
en el final. Por ejemplo <strong class="userinput"><code>1:2/5:3</code></strong> hace lo correcto y <strong
class="userinput"><code>float(1:2/5:3)</code></strong> incluso le da los números de punto flotante y es
ligeramente más precisa que <strong class="userinput"><code>1,0:0,4:3,0</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Crea un número imaginario
(multiplicando <code class="varname">a</code> por el imaginario). Tenga en cuenta que normalmente el
número<code class="varname">i</code> se escribe <strong class="userinput"><code>1i</code></strong>. De modo
que lo descrito arriba es equivalente a </p><pre class="programlisting">(a)*1i
- </pre></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Escapa un identificador de modo que no sea
evaluado. O escapa una matriz de modo que no sea expandida.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Intercambia el valor de <code
class="varname">a</code> con el valor de <code class="varname">b</code>. Actualmente no funciona con rangos
de elementos matriciales. Devuelve <code class="constant">null</code>. Está disponible desde la versión
1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment
a</code></strong></span></dt><dd><p>Incrementa la variable <code class="varname">a</code> en 1. Si <code
class="varname">a</code> es una matriz entonces incrementará cada uno de los elementos. Es equivalente a
<strong class="userinput"><code>a=a+1</code></strong> pero más rápido. Devuelve <code
class="constant">null</code>. Está disponi
ble desde la versión 1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment a by
b</code></strong></span></dt><dd><p>Incrementa la variable <code class="varname">a</code> en <code
class="varname">b</code>. Si <code class="varname">a</code> es una matriz, entonces incrementa cada elemento.
Es equivalente a <strong class="userinput"><code>a=a+b</code></strong>, pero más rápido. Devuelve null <code
class="constant">null</code>. Está disponible desde la versión 1.0.13.</p></dd></dl></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Nota</h3><p>El operador @() hace el
operador «:» más útil. Con éste puede especificar regiones dentro de una matriz. De modo que a@(2:4,6)
representa las filas 2, 3 y 4 de la columna 6. O @(,1:2) devuelve las dos primeras columnas de una matriz.
Puede asignar al operador @() siempre que el valor sea una matriz cuyo tamaño coincida con el tamaño de la
región asignada o cualq
uier otro tipo de valor.</p></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Nota</h3><p>Los operadores de comparación (excepto el operador <=> que se comporta de un
modo normal), no son estrictamente operadores binarios, de hecho pueden agruparse de una forma matemática
estándar, por ejemplo: (1<x<=y<5) es una expresión booleana válida y significa lo que debería, es
decir, (1<x and x≤y and y<5)</p></div><div class="note" style="margin-left: 0.5in; margin-right:
0.5in;"><h3 class="title">Nota</h3><p>El operador unario «menos» opera de un modo distinto dependiendo del
lugar donde aparece. Si lo hace antes de un número su prioridad es muy alta. Si aparece delante de una
expresión tendrá menos prioridad que los operadores potencia y factorial. De este modo, por ejemplo, <strong
class="userinput"><code>-1^k</code></strong> es en realidad <strong
class="userinput"><code>(-1)^k</code></strong>, sin embargo
<strong class="userinput"><code>-foo(1)^k</code></strong> es realmente <strong
class="userinput"><code>-(foo(1)^k)</code></strong>. Por lo tanto, tenga cuidado con el uso de este operador
y si tiene alguna duda, use paréntesis.</p></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch05s06.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch05.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Evaluación modular
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%"
align="right" valign="top"> Capítulo 6. Programar con GEL</td></tr></table></div></body></html>
+</pre><p>Cuando los números implicados son números en coma flotante, por ejemplo <strong
class="userinput"><code>1.0:0.4:3.0</code></strong>, la salida es lo que se espera a pesar de la adición de
0,4 a 1,0 cinco veces es en realidad sólo un poco más de 3,0 debido a la forma en que los números de coma
flotante se almacenan en la base 2 (no hay 0.4, el número real almacenado es sólo ligeramente más grande). La
forma en que se maneja es el mismo que en los bucles «for», «sum», y «prod». Si el final está dentro de
<strong class="userinput"><code>2^-20</code></strong> veces el tamaño de paso del punto final, se utiliza el
punto final y suponemos que no eran errores de redondeo. Esto no es perfecto, pero maneja la mayoría de los
casos. Esta comprobación se realiza sólo desde la versión 1.0.18 en adelante, así que la ejecución de su
código puede ser diferente en las versiones anteriores. Si quiere evitar este problema, utilice los números
racionales reales
, posiblemente usando el <code class="function">float</code> si quiere obtener los números de punto flotante
en el final. Por ejemplo <strong class="userinput"><code>1:2/5:3</code></strong> hace lo correcto y <strong
class="userinput"><code>float(1:2/5:3)</code></strong> incluso le da los números de punto flotante y es
ligeramente más precisa que <strong class="userinput"><code>1,0:0,4:3,0</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
+ written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
+ </p><pre class="programlisting">(a)*1i
+ </pre><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Escapa un identificador de modo que no sea
evaluado. O escapa una matriz de modo que no sea expandida.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Intercambia el valor de <code
class="varname">a</code> con el valor de <code class="varname">b</code>. Actualmente no funciona con rangos
de elementos matriciales. Devuelve <code class="constant">null</code>. Está disponible desde la versión
1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment
a</code></strong></span></dt><dd><p>Incrementa la variable <code class="varname">a</code> en 1. Si <code
class="varname">a</code> es una matriz entonces incrementará cada uno de los elementos. Es equivalente a
<strong class="userinput"><code>a=a+1</code></strong> pero más rápido. Devuelve <code
class="constant">null</code>. Está disp
onible desde la versión 1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment a
by b</code></strong></span></dt><dd><p>Incrementa la variable <code class="varname">a</code> en <code
class="varname">b</code>. Si <code class="varname">a</code> es una matriz, entonces incrementa cada elemento.
Es equivalente a <strong class="userinput"><code>a=a+b</code></strong>, pero más rápido. Devuelve null <code
class="constant">null</code>. Está disponible desde la versión 1.0.13.</p></dd></dl></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Nota</h3><p>El operador @() hace el
operador «:» más útil. Con éste puede especificar regiones dentro de una matriz. De modo que a@(2:4,6)
representa las filas 2, 3 y 4 de la columna 6. O @(,1:2) devuelve las dos primeras columnas de una matriz.
Puede asignar al operador @() siempre que el valor sea una matriz cuyo tamaño coincida con el tamaño de la
región asignada o cu
alquier otro tipo de valor.</p></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Nota</h3><p>Los operadores de comparación (excepto el operador <=> que se comporta de un
modo normal), no son estrictamente operadores binarios, de hecho pueden agruparse de una forma matemática
estándar, por ejemplo: (1<x<=y<5) es una expresión booleana válida y significa lo que debería, es
decir, (1<x and x≤y and y<5)</p></div><div class="note" style="margin-left: 0.5in; margin-right:
0.5in;"><h3 class="title">Nota</h3><p>El operador unario «menos» opera de un modo distinto dependiendo del
lugar donde aparece. Si lo hace antes de un número su prioridad es muy alta. Si aparece delante de una
expresión tendrá menos prioridad que los operadores potencia y factorial. De este modo, por ejemplo, <strong
class="userinput"><code>-1^k</code></strong> es en realidad <strong
class="userinput"><code>(-1)^k</code></strong>, sin emba
rgo <strong class="userinput"><code>-foo(1)^k</code></strong> es realmente <strong
class="userinput"><code>-(foo(1)^k)</code></strong>. Por lo tanto, tenga cuidado con el uso de este operador
y si tiene alguna duda, use paréntesis.</p></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch05s06.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch05.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Evaluación modular
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%"
align="right" valign="top"> Capítulo 6. Programar con GEL</td></tr></table></div></body></html>
diff --git a/help/es/html/ch06s05.html b/help/es/html/ch06s05.html
index 09cdffd..915714c 100644
--- a/help/es/html/ch06s05.html
+++ b/help/es/html/ch06s05.html
@@ -1,4 +1,12 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Variables globales y
ámbito de variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch06.html" title="Capítulo 6. Programar con
GEL"><link rel="prev" href="ch06s04.html" title="Operadores de comparación"><link rel="next"
href="ch06s06.html" title="Variables de parámetros"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Variables globales y ámbito de variables</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Anterior</a> </td><th width="60%"
align="center">Capítulo 6. Programar con GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Siguiente</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Variables globales y ámbito de variables</h2></div></div></div><p>GEL
es un <a class="ulink" href="http://es.wikipedia.org/wiki/%C3%81mbito_(programaci%C3%B3n)"
target="_top">lenguaje con ámbitos dinámicos</a>. Esto se explicará más adelante. Esto significa que a las
variables ordinarias y a las funciones se les asigna un ámbito de manera dinámica. La única excepción son las
<a class="link" href="ch06s06.html" title="Variables de parámetros">variables de parámetros</a>, que siempre
son globales.</p><p>Al igual que la mayoría de los lenguajes de programación, GEL tiene diferentes tipos de
variables. Normalmente, cuando se define una variable en una función ésta es visible desde esa función y
desde todas las funciones que se llamen (todos los contextos superiores). Por ejemplo, suponga que una
función <code class="function">f</code> de
fine una variable <code class="varname">a</code> y luego llama a otra función <code
class="function">g</code>. Entonces, la función <code class="function">g</code> puede hacer referencia a la
variable <code class="varname">a</code>. Pero, una vez que la ejecución de <code class="function">f</code>
concluye, la variable <code class="varname">a</code> sale del ámbito. Por ejemplo, el siguiente código
imprime el número 5. No se puede llamar a la función <code class="function">g</code> desde el nivel más alto
(fuera de <code class="function">f</code>, dado que <code class="varname">a</code> no se habrá
definido).</p><p>Si define una variable dentro de una función, ésta anulará toda variable definida al llamar
a funciones. Por ejemplo, si modifica el código anterior y escribe: </p><pre class="programlisting">function
f() = (a:=5; g());
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Variables globales y
ámbito de variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch06.html" title="Capítulo 6. Programar con
GEL"><link rel="prev" href="ch06s04.html" title="Operadores de comparación"><link rel="next"
href="ch06s06.html" title="Variables de parámetros"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Variables globales y ámbito de variables</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Anterior</a> </td><th width="60%"
align="center">Capítulo 6. Programar con GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Siguiente</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Variables globales y ámbito de variables</h2></div></div></div><p>
+ GEL is a
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ dynamically scoped language</a>. We will explain what this
+ means below. That is, normal variables and functions are dynamically
+ scoped. The exception are
+ <a class="link" href="ch06s06.html" title="Variables de parámetros">parameter variables</a>,
+ which are always global.
+ </p><p>Al igual que la mayoría de los lenguajes de programación, GEL tiene diferentes tipos de
variables. Normalmente, cuando se define una variable en una función ésta es visible desde esa función y
desde todas las funciones que se llamen (todos los contextos superiores). Por ejemplo, suponga que una
función <code class="function">f</code> define una variable <code class="varname">a</code> y luego llama a
otra función <code class="function">g</code>. Entonces, la función <code class="function">g</code> puede
hacer referencia a la variable <code class="varname">a</code>. Pero, una vez que la ejecución de <code
class="function">f</code> concluye, la variable <code class="varname">a</code> sale del ámbito. Por ejemplo,
el siguiente código imprime el número 5. No se puede llamar a la función <code class="function">g</code>
desde el nivel más alto (fuera de <code class="function">f</code>, dado que <code class="varname">a</code> no
se habrá definido).</p><p>Si de
fine una variable dentro de una función, ésta anulará toda variable definida al llamar a funciones. Por
ejemplo, si modifica el código anterior y escribe: </p><pre class="programlisting">function f() = (a:=5; g());
function g() = print(a);
a:=10;
f();
diff --git a/help/es/html/ch07s02.html b/help/es/html/ch07s02.html
index bf71fbb..ad5f1a7 100644
--- a/help/es/html/ch07s02.html
+++ b/help/es/html/ch07s02.html
@@ -1,4 +1,35 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Sintaxis de nivel
superior</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch07.html" title="Capítulo 7. Programación
avanzada con GEL"><link rel="prev" href="ch07.html" title="Capítulo 7. Programación avanzada con GEL"><link
rel="next" href="ch07s03.html" title="Devolver funciones"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Sintaxis de nivel superior</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch07.html">Anterior</a> </td><th width="60%" align="center">Capítulo 7.
Programación avanzada con GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch07s03.html">Siguiente</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-toplevel-syntax"></a>Sintaxis de nivel superior</h2></div></div></div><p>Cuando se indroduce
una sentencia en el nivel más alto, la sintaxis es distinta a la que se utiliza cuando se introduce entre
paréntesis o dentro de una función. En el nivel más alto la tecla «Intro» tiene el mismo efecto que al
pulsarla en la línea de comandos. Piense en un programa como una secuencia de líneas introducidas en la línea
de comandos. En particular, no necesita introducir el separador al final de la línea (salvo que sea parte de
varias sentencias dentro de paréntesis).</p><p>El siguiente código, aunque funcione bien en la función, puede
producir un error al introducirlo en el nivel más alto de un programa. </p><pre class="programlisting">if
Algo() then
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Sintaxis de nivel
superior</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch07.html" title="Capítulo 7. Programación
avanzada con GEL"><link rel="prev" href="ch07.html" title="Capítulo 7. Programación avanzada con GEL"><link
rel="next" href="ch07s03.html" title="Devolver funciones"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Sintaxis de nivel superior</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch07.html">Anterior</a> </td><th width="60%" align="center">Capítulo 7.
Programación avanzada con GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch07s03.html">Siguiente</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-toplevel-syntax"></a>Sintaxis de nivel superior</h2></div></div></div><p>
+ The syntax is slightly different if you enter statements on
+ the top level versus when they are inside parentheses or
+ inside functions. On the top level, enter acts the same as if
+ you press return on the command line. Therefore think of programs
+ as just a sequence of lines as if they were entered on the command line.
+ In particular, you do not need to enter the separator at the end of the
+ line (unless it is of course part of several statements inside
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
+ </p><p>El siguiente código, aunque funcione bien en la función, puede producir un error al
introducirlo en el nivel más alto de un programa. </p><pre class="programlisting">if Algo() then
HacerAlgo()
else
HacerOtraCosa()
diff --git a/help/es/html/ch11s04.html b/help/es/html/ch11s04.html
index 74f5c42..6f2a78b 100644
--- a/help/es/html/ch11s04.html
+++ b/help/es/html/ch11s04.html
@@ -1 +1,28 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Constantes</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s03.html" title="Parámetros"><link
rel="next" href="ch11s05.html" title="Numérico"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Constantes</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s03.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-constants"></a>Constantes</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Constante de Catalan, aproximadamente 0,915... Se define para las
series donde los términos son <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, donde <code
class="varname">k</code> tiene un rango desde 0 a infinito.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Catalan%27s_constant"; target="_top">Wikipedia</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/CatalansConstant.html"; target="_top">la enciclopedia matemática
Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alias: <code class="function">gamma</c
ode></p><p>Constante gamma de Euler. También llamada constante de Euler-Mascheroni.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html"; target="_top">Mathworld</a> para obtener
más información.</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>El
número áureo.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/GoldenRatio";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a name="gel-f
unction-Gravity"></a>Gravity</span></dt><dd><pre class="synopsis">Gravedad</pre><p>La aceleración en caída
libre al nivel del mar en metros por segundos al cuadrado. Es la constante de gravedad estandarizada y su
valor es 9.80665. La gravedad en un desfiladero de un bosque es diferente debido principalmente a la
diferencia de altitud y al hecho de que la Tierra no es perfectamente redonda ni uniforme.</p><p>Consulte la
<a class="ulink" href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a> para
obtener más información.</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>La base del logaritmo natural. <strong class="userinput"><code>e^x</code></strong>
es la función exponencial <a class="link" href="ch11s05.html#gel-function-exp"><code
class="function">exp</code></a>. Su valor es aproximadamente 2.71828182846... Este número se llama número de
Euler, aúnque hay varios números que se lla
man también Euler. Un ejemplo es la constante gamma: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">Constante de
Euler</code></a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/E_(mathematical_constant)" target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/E"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>El número pi, que es la relación de la circunferencia de un círculo con su
diámetro. Esto es aproximadamente 3,14159265359...</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/Pi"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/Pi.html"; targe
t="_top">Mathworld</a> para obtener más información.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s03.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Parámetros </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%" align="right"
valign="top"> Numérico</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Constantes</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s03.html" title="Parámetros"><link
rel="next" href="ch11s05.html" title="Numérico"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Constantes</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s03.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-constants"></a>Constantes</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Constante de Catalan, aproximadamente 0,915... Se define para las
series donde los términos son <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, donde <code
class="varname">k</code> tiene un rango desde 0 a infinito.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alias: <code class="function">gamma</code></p><p>Constante gamma de
Euler. También llamada constante de Euler-Mascheroni.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>El
número áureo.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravedad</pre><p>La aceleración en caída libre al nivel del mar en metros por segundos al
cuadrado. Es la constante de gravedad estandarizada y su valor es 9.80665. La gravedad en un desfiladero de
un bosque es diferente debido principalmente a la diferencia de altitud y al hecho de que la Tierra no es
perfectamente redonda ni uniforme.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>La base del logaritmo natural. <strong class="userinput"><code>e^x</code></strong>
es la función exponencial <a class="link" href="ch11s05.html#gel-function-exp"><code
class="function">exp</code></a>. Su valor es aproximadamente 2.71828182846... Este número se llama número de
Euler, aúnque hay varios números que se llaman también Euler. Un ejemplo es la constante gamma: <a
class="link" href="ch11s04.html#gel-function-EulerConstant"><code class="function">Constante de
Euler</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>El número pi, que es la relación de la circunferencia de un círculo con su
diámetro. Esto es aproximadamente 3,14159265359...</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s05.html">Siguiente</a></td></tr><tr><td width="40%" align="left"
valign="top">Parámetros </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top">
Numérico</td></tr></table></div></body></html>
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-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Numérico</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manual
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alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Numérico</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s04.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de funciones
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="cle
ar: both"><a name="genius-gel-function-list-numeric"></a>Numérico</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Alias: <code class="function">abs</code></p><p>Valor absoluto de un número y, si <code
class="varname">x</code> es un valor complejo, el módulo de <code class="varname">x</code>. Es decir, es la
distancia entre <code class="varname">x</code> y el origen. Esto es equivalente a <strong
class="userinput"><code>|x|</code></strong>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (valor absoluto)</a>, <a class="ulink"
href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath (módulo)</a>, <a class="ulink"
href="http://mathworld.wolf
ram.com/AbsoluteValue.html" target="_top">Mathworld (valor absoluto)</a> o <a class="ulink"
href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld (módulo complejo)</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Chop"></a>Chop</span></dt><dd><pre class="synopsis">Chop (x)</pre><p>Reemplazar números
muy pequeños por cero.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alias: <code class="function">conj</code><code
class="function">Conj</code></p><p>Calcula el conjugado complejo del número complejo <code
class="varname">z</code>. Si <code class="varname">z</code> es un vector o una matriz, se conjugan todos sus
elementos.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="ter
m"><a name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Obtener el denominador de un número racional.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Devolver la parte fraccional de un número.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alias: <code class="function">ImaginaryPart</code></p><p>Obtener la parte
imaginaria de un número complejo. Por ejemplo <strong class="userinput"><code>Re(3+4i)</code></strong>
yields 4.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>División sin resto.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Comprueba si el argumento es un número complejo (no real). Observe que hacemos énfasis en
número no real. Es decir, <strong class="userinput"><code>IsComplex(3)</code></strong> que devuelve «false»,
mientras que <strong class="userinput"><code>IsComplex(3-1i)</code></strong> devuelve
«true».</p></dd><dt><span class="term"><a
name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Comprobar si el argumento es, posiblemente, un número
racional complejo. Esto es, si tanto la p
arte real como la imaginaria se dan como números racionales. Por supuesto, racional significa simplemente
que «no se almacena como un número en coma flotante».</p></dd><dt><span class="term"><a
name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre class="synopsis">IsFloat (num)</pre><p>Comprobar
si el argumento es un número real en coma flotante (no complejo).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(num)</pre><p>Alias: <code class="function">IsComplexInteger</code></p><p>Comprueba si un argumento es un
posible número entero complejo. Es decir, un entero complejo es un número de la forma <strong
class="userinput"><code>n+1i*m</code></strong> donde <code class="varname">n</code> y <code
class="varname">m</code> son enteros.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger (num)</pre><p>Com
probar si el argumento es un entero (no complejo).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Comprobar si el argumento es un entero real no negativo.
Esto es, cualquier número entero positivo o cero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Alias: <code
class="function">IsNaturalNumber</code></p><p>Comprueba si el argumento es un entero real positivo. Tenga en
cuenta que se acepta el convenio de que 0 no es un número natural.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Comprobar si el argumento es un número racional (no complejo). Por supuesto, racional significa
«no almacenado como un número en coma flotante».</p></dd><dt><s
pan class="term"><a name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal
(num)</pre><p>Comprobar si el argumento es un número real</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator
(x)</pre><p>Obtener el numerador de un número racional.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Alias: <code class="function">RealPart</code></p><p>Obtiene la parte real de
un número complejo. Por ejemplo <strong class="userinput"><code>Re(3+4i)</code></strong> devuelve
3.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Real_part";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Sign"></a>Sign
</span></dt><dd><pre class="synopsis">Sign (x)</pre><p>Alias: <code
class="function">sign</code></p><p>Devolver el signo de un número. Devuelve <code class="literal">-1</code>
si es negativo, <code class="literal">0</code> si es cero y <code class="literal">1</code> si es positivo. Si
<code class="varname">x</code> es un valor complejo <code class="function">Sign</code> devuelve su dirección
o 0.</p></dd><dt><span class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre
class="synopsis">ceil (x)</pre><p>Alias: <code class="function">Ceiling</code></p><p>Obtener el menor número
entero mayor o igual a <code class="varname">n</code>. Ejemplos: </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>ceil(1,1)</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Numérico</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manual
de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de funciones GEL"><link rel="prev"
href="ch11s04.html" title="Constantes"><link rel="next" href="ch11s06.html"
title="Trigonometría"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Numérico</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s04.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de funciones
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="cle
ar: both"><a name="genius-gel-function-list-numeric"></a>Numérico</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Alias: <code class="function">abs</code></p><p>Valor absoluto de un número y, si <code
class="varname">x</code> es un valor complejo, el módulo de <code class="varname">x</code>. Es decir, es la
distancia entre <code class="varname">x</code> y el origen. Esto es equivalente a <strong
class="userinput"><code>|x|</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
+ <a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld
(complex modulus)</a>
+for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Reemplazar números muy pequeños por cero.</p></dd><dt><span
class="term"><a name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alias: <code class="function">conj</code><code
class="function">Conj</code></p><p>Calcula el conjugado complejo del número complejo <code
class="varname">z</code>. Si <code class="varname">z</code> es un vector o una matriz, se conjugan todos sus
elementos.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Obtener el denominador de un número racional.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Devolver la parte fraccional de un número.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alias: <code class="function">ImaginaryPart</code></p><p>Obtener la parte
imaginaria de un número complejo. Por ejemplo <strong class="userinput"><code>Re(3+4i)</code></strong>
yields 4.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>División sin resto.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Comprueba si el argumento es un número complejo (no real). Observe que hacemos énfasis en
número no real. Es decir, <strong class="userinput"><code>IsComplex(3)</code></strong> que devuelve «false»,
mientras que <strong class="userinput"><code>IsComplex(3-1i)</code></strong> devuelve
«true».</p></dd><dt><span class="term"><a
name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Comprobar si el argumento es, posiblemente, un número
racional complejo. Esto es, si tanto la parte real como la imaginaria se dan como números racionales. Por
supuesto, racional significa simple
mente que «no se almacena como un número en coma flotante».</p></dd><dt><span class="term"><a
name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre class="synopsis">IsFloat (num)</pre><p>Comprobar
si el argumento es un número real en coma flotante (no complejo).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(num)</pre><p>Alias: <code class="function">IsComplexInteger</code></p><p>Comprueba si un argumento es un
posible número entero complejo. Es decir, un entero complejo es un número de la forma <strong
class="userinput"><code>n+1i*m</code></strong> donde <code class="varname">n</code> y <code
class="varname">m</code> son enteros.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Comprobar si el argumento es un entero (no complejo).</p></dd><dt><span class="term"><a
name="gel-functi
on-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre class="synopsis">IsNonNegativeInteger
(num)</pre><p>Comprobar si el argumento es un entero real no negativo. Esto es, cualquier número entero
positivo o cero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Alias: <code
class="function">IsNaturalNumber</code></p><p>Comprueba si el argumento es un entero real positivo. Tenga en
cuenta que se acepta el convenio de que 0 no es un número natural.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Comprobar si el argumento es un número racional (no complejo). Por supuesto, racional significa
«no almacenado como un número en coma flotante».</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal
(num)</pre><p>Comprobar si el argumento es un número real</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator
(x)</pre><p>Obtener el numerador de un número racional.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Alias: <code class="function">RealPart</code></p><p>Obtiene la parte real de
un número complejo. Por ejemplo <strong class="userinput"><code>Re(3+4i)</code></strong> devuelve 3.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Alias: <code class="function">sign</code></p><p>Devolver el signo de un
número. Devuelve <code class="literal">-1</code> si es negativo, <code class="literal">0</code> si es cero y
<code class="literal">1</code> si es positivo. Si <code class="varname">x</code> es un valor complejo <code
class="function">Sign</code> devuelve su dirección o 0.</p></dd><dt><span class="term"><a
name="gel-function-ceil"></a>ceil</span></dt><dd><pre class="synopsis">ceil (x)</pre><p>Alias: <code
class="function">Ceiling</code></p><p>Obtener el menor número entero mayor o igual a <code
class="varname">n</code>. Ejemplos: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ceil(1,1)</code></strong>
= 2
<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1,1)</code></strong>
= -1
-</pre><p>Tenga en cuenta que los números en coma flotante se almacenan en binario y que puede que el
resultado no sea lo que espera. Por ejemplo <strong class="userinput"><code>ceil(420/4.2)</code></strong>
devuelve 101 en vez de 100. Esto sucede porque en realidad 4,2 es ligeramente menor que 4,2. Utilice la
representación racional <strong class="userinput"><code>42/10</code></strong> si quiere exactitud
aritmética.</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>La función exponencial. Esto es la función <strong
class="userinput"><code>e^x</code></strong> donde <code class="varname">e</code> es la <a class="link"
href="ch11s04.html#gel-function-e">base del logaritmo natural</a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Exponential_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> o <a cla
ss="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html"; target="_top">Mathworld</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-float"></a>float</span></dt><dd><pre class="synopsis">float (x)</pre><p>Convertir un
número en un valor en coma flotante. Esto devuelve la representación en coma flotante del número <code
class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Alias: <code
class="function">Floor</code></p><p>Obtener el entero más alto menor o igual que <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>El logaritmo natural, logaritmo en base <code
class="varname">e</code>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Natural_logarithm"; target="_top">Wikipedia</a> o <a class="ulink"
href="http:
//planetmath.org/LogarithmFunction" target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/NaturalLogarithm.html"; target="_top">Mathworld</a> para más
información.</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logaritmo de <code
class="varname">x</code> en base <code class="varname">b</code> (se llama <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> en modo módulo),
si no se indica la base, se utiliza <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logaritmo de <code
class="varname">x</code> en base 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pr
e><p>Alias: <code class="function">lg</code></p><p>Logaritmo de <code class="varname">x</code> en base
2.</p></dd><dt><span class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synopsis">max
(a,args...)</pre><p>Alias: <code class="function">Max</code><code
class="function">Maximum</code></p><p>Devuelve el máximo de los argumentos o las matrices.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,args...)</pre><p>Alias: <code class="function">Min</code><code
class="function">Minimum</code></p><p>Devuelve el mínimo de los argumentos o las matrices.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(tamaño...)</pre><p>Generar valores en coma flotante aleatorios en el rango <code
class="literal">[0,1)</code>. Si se indica «tamaño», entonces devuelve una matriz (si se especifican dos
números) o un vector (si se especifica un número
).</p></dd><dt><span class="term"><a name="gel-function-randint"></a>randint</span></dt><dd><pre
class="synopsis">randint (máx,tamaño...)</pre><p>Generar número enteros aleatorios en el rango <code
class="literal">[0,máx)</code>. Si se indica «tamaño», entonces devuelve una matriz (si se especifican dos
números) o un vector (si se especifica un número). Por ejemplo, </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>randint(4)</code></strong>
+</pre><p>Tenga en cuenta que los números en coma flotante se almacenan en binario y que puede que el
resultado no sea lo que espera. Por ejemplo <strong class="userinput"><code>ceil(420/4.2)</code></strong>
devuelve 101 en vez de 100. Esto sucede porque en realidad 4,2 es ligeramente menor que 4,2. Utilice la
representación racional <strong class="userinput"><code>42/10</code></strong> si quiere exactitud
aritmética.</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>La función exponencial. Esto es la función <strong
class="userinput"><code>e^x</code></strong> donde <code class="varname">e</code> es la <a class="link"
href="ch11s04.html#gel-function-e">base del logaritmo natural</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Convertir un número en un valor en coma flotante. Esto devuelve la
representación en coma flotante del número <code class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Alias: <code
class="function">Floor</code></p><p>Obtener el entero más alto menor o igual que <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>El logaritmo natural, logaritmo en base <code class="varname">e</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logaritmo de <code
class="varname">x</code> en base <code class="varname">b</code> (se llama <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> en modo módulo),
si no se indica la base, se utiliza <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logaritmo de <code
class="varname">x</code> en base 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Alias: <code
class="function">lg</code></p><p>Logaritmo de <code class="varname">x</code> en base 2.</p></dd><dt><span
class="term"><a name="gel-function-max"></a>max</span></dt><dd><pr
e class="synopsis">max (a,args...)</pre><p>Alias: <code class="function">Max</code><code
class="function">Maximum</code></p><p>Devuelve el máximo de los argumentos o las matrices.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,args...)</pre><p>Alias: <code class="function">Min</code><code
class="function">Minimum</code></p><p>Devuelve el mínimo de los argumentos o las matrices.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(tamaño...)</pre><p>Generar valores en coma flotante aleatorios en el rango <code
class="literal">[0,1)</code>. Si se indica «tamaño», entonces devuelve una matriz (si se especifican dos
números) o un vector (si se especifica un número).</p></dd><dt><span class="term"><a
name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(máx,tamaño...)</pre><p>Generar número enteros aleatorios en el r
ango <code class="literal">[0,máx)</code>. Si se indica «tamaño», entonces devuelve una matriz (si se
especifican dos números) o un vector (si se especifica un número). Por ejemplo, </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>randint(4)</code></strong>
= 3
<code class="prompt">genius></code> <strong class="userinput"><code>randint(4,2)</code></strong>
=
diff --git a/help/es/html/ch11s06.html b/help/es/html/ch11s06.html
index 0d48b03..00e648d 100644
--- a/help/es/html/ch11s06.html
+++ b/help/es/html/ch11s06.html
@@ -1,2 +1,34 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometría</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s05.html" title="Numérico"><link
rel="next" href="ch11s07.html" title="Teoría de números"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometría</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="t
itle" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonometría</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alias: <code
class="function">arccos</code></p><p>La función arccos (inversa del cos).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Alias: <code
class="function">arccosh</code></p><p>La función arccosh (inversa del cosh).</p></dd><dt><span
class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot
(x)</pre><p>Alias: <code class="function">arccot</code></p><p>La función arccot (inversa de la
cot)</p></dd><dt><span class="term"><a name="gel-function-acoth"></a>acoth</span></dt><dd><pre
class="synopsis">acoth (x)</pre><p>Alias: <code class="function">arccoth</code></p><p>La función a
rccoth (inversa de la coth).</p></dd><dt><span class="term"><a
name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc (x)</pre><p>Alias: <code
class="function">arccsc</code></p><p>La inversa de la función cosecante.</p></dd><dt><span class="term"><a
name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch (x)</pre><p>Alias: <code
class="function">arccsch</code></p><p>La inversa de la función cosecante hiperbólica.</p></dd><dt><span
class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec
(x)</pre><p>Alias: <code class="function">arcsec</code></p><p>La inversa de la función
secante.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Alias: <code class="function">arcsech</code></p><p>La inversa de la
función secante hiperbólica.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre
class="synopsis">asin (x)</pre><p>Alias: <code class="function">arcsin</code></p><p>La función arcsen
(inversa del sen).</p></dd><dt><span class="term"><a name="gel-function-asinh"></a>asinh</span></dt><dd><pre
class="synopsis">asinh (x)</pre><p>Alias: <code class="function">arcsinh</code></p><p>La función arcsenh
(inversa del senh).</p></dd><dt><span class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre
class="synopsis">atan (x)</pre><p>Alias: <code class="function">arctan</code></p><p>Calcula la función
«arctan» (inversa de «tan»).</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alias: <code class="function">arctanh</code></p><p>La
función arctanh (inversa de la tanh).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alias: <code
class="function">arctan2</code></p><p>Calcula la función «arctan2». Si <strong
class="userinput"><code>x>0</code></strong>, entonces devuelve <strong
class="userinput"><code>atan(y/x)</code></strong>. Si <strong class="userinput"><code>x<0</code></strong>,
entonces devuelve <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>. Cuando <strong
class="userinput"><code>x=0</code></strong> devuelve <strong class="userinput"><code>sign(y) *
- pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> devuelve 0 en
lugar de fallar.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2";
target="_top">Wikipedia</a> o <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-cos"></a>cos</span></dt><dd><pre class="synopsis">cos (x)</pre><p>Calcula la función
coseno.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-cosh"></a>cosh</span></dt><dd><pre class="synopsis">cosh (x)</pre><p>Calcula la función
coseno hiperbólico.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.o
rg/wiki/Hyperbolic_function" target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>La función cotangente.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>La función cotangente hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más infor
mación.</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>La función cosecante.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>La función cosecante hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>La función secante.</p><p>Consulte la <a class="u
link" href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-sech"></a>sech</span></dt><dd><pre class="synopsis">sech (x)</pre><p>La función secante
hiperbólica.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_function";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-sin"></a>sin</span></dt><dd><pre class="synopsis">sin (x)</pre><p>Calcula la función
seno.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_t
op">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-sinh"></a>sinh</span></dt><dd><pre class="synopsis">sinh (x)</pre><p>Calcula la función
seno hiperbólico.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calcula la función tangente.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre
<p>La función tangente hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s05.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td width="40%" align="right">
<a accesskey="n" href="ch11s07.html">Siguiente</a></td></tr><tr><td width="40%" align="left"
valign="top">Numérico </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top"> Teoría de
números</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometría</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s05.html" title="Numérico"><link
rel="next" href="ch11s07.html" title="Teoría de números"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometría</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="t
itle" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonometría</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alias: <code
class="function">arccos</code></p><p>La función arccos (inversa del cos).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Alias: <code
class="function">arccosh</code></p><p>La función arccosh (inversa del cosh).</p></dd><dt><span
class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot
(x)</pre><p>Alias: <code class="function">arccot</code></p><p>La función arccot (inversa de la
cot)</p></dd><dt><span class="term"><a name="gel-function-acoth"></a>acoth</span></dt><dd><pre
class="synopsis">acoth (x)</pre><p>Alias: <code class="function">arccoth</code></p><p>La función a
rccoth (inversa de la coth).</p></dd><dt><span class="term"><a
name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc (x)</pre><p>Alias: <code
class="function">arccsc</code></p><p>La inversa de la función cosecante.</p></dd><dt><span class="term"><a
name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch (x)</pre><p>Alias: <code
class="function">arccsch</code></p><p>La inversa de la función cosecante hiperbólica.</p></dd><dt><span
class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec
(x)</pre><p>Alias: <code class="function">arcsec</code></p><p>La inversa de la función
secante.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Alias: <code class="function">arcsech</code></p><p>La inversa de la
función secante hiperbólica.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre
class="synopsis">asin (x)</pre><p>Alias: <code class="function">arcsin</code></p><p>La función arcsen
(inversa del sen).</p></dd><dt><span class="term"><a name="gel-function-asinh"></a>asinh</span></dt><dd><pre
class="synopsis">asinh (x)</pre><p>Alias: <code class="function">arcsinh</code></p><p>La función arcsenh
(inversa del senh).</p></dd><dt><span class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre
class="synopsis">atan (x)</pre><p>Alias: <code class="function">arctan</code></p><p>Calcula la función
«arctan» (inversa de «tan»).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alias: <code class="function">arctanh</code></p><p>La función arctanh
(inversa de la tanh).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alias: <code
class="function">arctan2</code></p><p>Calcula la función «arctan2». Si <strong
class="userinput"><code>x>0</code></strong>, entonces devuelve <strong
class="userinput"><code>atan(y/x)</code></strong>. Si <strong class="userinput"><code>x<0</code></strong>,
entonces devuelve <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>. Cuando <strong
class="userinput"><code>x=0</code></strong> devuelve <strong class="userinput"><code>sign(y) *
+ pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> devuelve 0 en
lugar de fallar.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Calcula la función coseno.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Calcula la función coseno hiperbólico.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>La función cotangente.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>La función cotangente hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>La función cosecante.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>La función cosecante hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>La función secante.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>La función secante hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Calcula la función seno.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Calcula la función seno hiperbólico.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calcula la función tangente.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>La función tangente hiperbólica.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> para obtener más
información.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s05.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s07.html">Siguiente</a></td></tr><tr><td width="40%" align="left"
valign="top">Numérico </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top"> Teoría de números</td></
tr></table></div></body></html>
diff --git a/help/es/html/ch11s07.html b/help/es/html/ch11s07.html
index 1c4a4ad..e328ff0 100644
--- a/help/es/html/ch11s07.html
+++ b/help/es/html/ch11s07.html
@@ -1,8 +1,71 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Teoría de
números</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de
funciones GEL"><link rel="prev" href="ch11s06.html" title="Trigonometría"><link rel="next"
href="ch11s08.html" title="Manipulación de matrices"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Teoría de números</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s06.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s08.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage">
<div><div><h2 class="title" style="clear: both"><a name="genius-gel-function-list-number-theory"></a>Teoría
de números</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span
class="term"><a name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>¿Son los números reales <code class="varname">a</code> and
<code class="varname">b</code> primos entre sí? Devuelve <code class="constant">true</code> o <code
class="constant">false</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Coprime_integers"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/RelativelyPrime.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-BernoulliNumber"></a>BernoulliNumber</span>
</dt><dd><pre class="synopsis">BernoulliNumber (n)</pre><p>Devolver el <code class="varname">n</code>-ésimo
número de Bernoulli.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> o <a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alias: <code class="function">CRT</code></p><p>Encontrar la
<code class="varname">x</code> que resuelve el sistema dado por el vector <code class="varname">a</code> y el
módulo de los elementos de <code class="varname">m</code>, utilizando el «teorema chino del
resto».</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/C
hineseRemainderTheorem" target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Dadas dos factorizaciones, dar la factorización del
producto.</p><p>Consulte la sección<a class="link"
href="ch11s07.html#gel-function-Factorize">factorizar</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Convertir un vector de valores mostrando potencias de b a un número.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Convertir un número en un vector de potencias para elementos en
base <co
de class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>Encontrar el logaritmo discreto de <code class="varname">n</code> en base <code
class="varname">b</code> en F<sub>q</sub>, el campo finito de orden <code class="varname">q</code>, donde
<code class="varname">q</code> es primo, utilizando el algoritmo de Silver-Pohlig-Hellman.</p><p>Consulte la
<a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/DiscreteLogarithm.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Comprueba la divisibilidad (si <code class="varname"
m</code> divide a <code class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi
(n)</pre><p>Calcular la función phi de Euler para <code class="varname">n</code>, que es el número de
enteros entre 1 y <code class="varname">n</code> primo relativo con <code
class="varname">n</code>.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/EulerPhifunction";
target="_top">Planetmath</a>, o <a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Devuelve <strong class="userinput"><code>n/d</code></strong> pero solo si <code
class="varname">d</code> es divisible ent
re <code class="varname">n</code>. Si <code class="varname">d</code> no es divisible entre <code
class="varname">n</code> entonces esta función devuelve basura. Esto es mucho mas rápido para números muy
grandes que la operación <strong class="userinput"><code>n/d</code></strong>, pero sólo es útil si se sabe
que la división es exacta.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize
(n)</pre><p>Devuelve la factorización de un número como una matriz. La primera fila son los números primos en
la factorización (incluido el 1) y la segunda fila son las potencias. Por ejemplo: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>Factorize(11*11*13)</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Teoría de
números</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de
funciones GEL"><link rel="prev" href="ch11s06.html" title="Trigonometría"><link rel="next"
href="ch11s08.html" title="Manipulación de matrices"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Teoría de números</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s06.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s08.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage">
<div><div><h2 class="title" style="clear: both"><a name="genius-gel-function-list-number-theory"></a>Teoría
de números</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span
class="term"><a name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>¿Son los números reales <code class="varname">a</code> and
<code class="varname">b</code> primos entre sí? Devuelve <code class="constant">true</code> o <code
class="constant">false</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Coprime_integers"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/RelativelyPrime.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-BernoulliNumber"></a>BernoulliNumber</span>
</dt><dd><pre class="synopsis">BernoulliNumber (n)</pre><p>Devolver el <code class="varname">n</code>-ésimo
número de Bernoulli.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alias: <code class="function">CRT</code></p><p>Encontrar la
<code class="varname">x</code> que resuelve el sistema dado por el vector <code class="varname">a</code> y el
módulo de los elementos de <code class="varname">m</code>, utilizando el «teorema chino del resto».</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Dadas dos factorizaciones, dar la factorización del
producto.</p><p>Consulte la sección<a class="link"
href="ch11s07.html#gel-function-Factorize">factorizar</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Convertir un vector de valores mostrando potencias de b a un número.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Convertir un número en un vector de potencias para elementos en
base <code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>Encontrar el loga
ritmo discreto de <code class="varname">n</code> en base <code class="varname">b</code> en F<sub>q</sub>, el
campo finito de orden <code class="varname">q</code>, donde <code class="varname">q</code> es primo,
utilizando el algoritmo de Silver-Pohlig-Hellman.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Comprueba la divisibilidad (si <code class="varname">m</code> divide a
<code class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>Calcular
la función phi de Euler para <code class="varname">n</code>, que es el número de enteros entre 1 y <code
class="varname">n</code> primo relativo con <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Devuelve <strong class="userinput"><code>n/d</code></strong> pero solo si <code
class="varname">d</code> es divisible entre <code class="varname">n</code>. Si <code class="varname">d</code>
no es divisible entre <code class="varname">n</code> entonces esta función devuelve basura. Esto es mucho mas
rápido para números muy grandes que la operación <strong class="userinput"><code>n/d</code></strong>, pero
sólo es útil si se sabe que la división es exacta.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize
(n)</pre><p>Devuelve la factorización de un número como una matriz. La primera fila son los números primos en
la factorización (incluido el 1) y la segunda fila son las potencias. Por ejemplo: </p><pre
class="screen"><code class="prompt">ge
nius></code> <strong class="userinput"><code>Factorize(11*11*13)</code></strong>
=
[1 11 13
- 1 2 1]</pre><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Factors"></a>Factors</span></dt><dd><pre class="synopsis">Factors (n)</pre><p>Devuelve
todos los factores de <code class="varname">n</code> en un vector. Esto incluye todos los factores no primos
como buenos. Incluye 1 y el mismo número. Así por ejemplo, para imprimir todos los números perfectos
(aquellos que son sumas de sus factores) hasta el número 1000 (esto es muy ineficiente) haga </p><pre
class="programlisting">for n=1 to 1000 do (
+ 1 2 1]</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>Devuelve todos los factores de <code class="varname">n</code> en un
vector. Esto incluye todos los factores no primos como buenos. Incluye 1 y el mismo número. Así por ejemplo,
para imprimir todos los números perfectos (aquellos que son sumas de sus factores) hasta el número 1000 (esto
es muy ineficiente) haga </p><pre class="programlisting">for n=1 to 1000 do (
if MatrixSum (Factors(n)) == 2*n then
print(n)
)
-</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,tries)</pre><p>Probar la factorización de Fermat de <code
class="varname">n</code> en <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, devuelve <code
class="varname">t</code> y <code class="varname">s</code> como un vector si es posible, <code
class="constant">null</code> de otra manera <code class="varname">tries</code> especifica el número de
intentos antes de abandonar </p><p>Es una buena factorización si su número es el producto de dos factores que
están muy cerca.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Encont
rar el primer elemento primitivo en F<sub>q</sub>, en el grupo de orden finito<code
class="varname">q</code>. Por supuesto, <code class="varname">q</code> debe de ser primo.</p></dd><dt><span
class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Encontrar un elemento primitivo aleatorio en
F<sub>q</sub>, en el grupo de orden finito <code class="varname">q</code> (q debe de ser
primo)</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Calcula la base del logaritmo discreto <code class="varname">b</code> de n en
F<sub>q</sub>, el grupo finito de orden <code class="varname">q</code> (<code class="varname">q</code> un
primo), utilizando el factor base <code class="varname">S</code>. <code class="varname">S</code> será una
columna de números primos y un
a segunda columna precalculada por <a class="link"
href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Ejecuta los pasos para los cálculos previos de
<a class="link" href="ch11s07.html#gel-function-IndexCalculus"><code
class="function">IndexCalculus</code></a> para logaritmos de base <code class="varname">b</code> en
F<sub>q</sub>, del grupo finito de orden <code class="varname">q</code> (<code class="varname">q</code> un
primo), para el factor base <code class="varname">S</code> (donde <code class="varname">S</code> es una
columna de vector de primos). Los registros se calculan previamente y se devuelven en la segunda
columna.</p></dd><dt><span class="term"><a name="gel-function-IsEven"></a>IsEven</span></dt><d
d><pre class="synopsis">IsEven (n)</pre><p>Comprueba si un entero es par.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Comprueba si un entero positivo <code
class="varname">p</code> es un exponente primo de Mersenne. Esto es si 2<sup>p</sup>-1 es un primo. Esto lo
hace mirando en una tabla de valores conocidos que es relativamente corta. Vea también <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> y <a class="link"
href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>, <a class="ulink"
href="http://mathworld.wolfram.com/MersennePrime.html"; target="_top">Mathworld</a> o <a class="ulin
k" href="http://www.mersenne.org/"; target="_top">GIMPS</a> para obtener más información.</p></dd><dt><span
class="term"><a name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Comprueba si un número racional <code class="varname">m</code> es una potencia <code
class="varname">n</code>-ésima perfecta. Consulte <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> y <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Comprueba su un
entero es impar.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Comprobar si un entero es una potencia perfecta, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>
IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare (n)</pre><p>Comprobar si un entero es
un cuadrado perfecto de un entero. El número será un entero real. Los enteros negativos, por supuesto, no son
perfectos cuadrados de enteros reales.</p></dd><dt><span class="term"><a
name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre class="synopsis">IsPrime (n)</pre><p>Comprueba si
dos números enteros son primos, para números menores que 2.5e10 la respuesta es determinista (si la hipótesis
de Riemann es verdadera). Para números más grandes, la probabilidad de un falso positivo depende de <a
class="link" href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. Significa que la probabilidad de un falso positivo es 1/4
de la potencia <code class="function">IsPrimeMillerRabinReps</code>. De manera predeterminada el valor de 22
produce una probabilidad entorno a 5.7e-14.</p><p>Si se devuelve <code clas
s="constant">false</code>, puede estar seguro de que el número es un compuesto. Si quiere estar totalmente
seguro de que tiene un número primo use <a class="link"
href="ch11s07.html#gel-function-MillerRabinTestSure"><code class="function">MillerRabinTestSure</code></a>
pero esto le puede llevar mucho más tiempo.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre class="synopsis">IsPrimitiveMod
(g,q)</pre><p>Comprobar si <code class="varname">g</code> es primario en F<sub>q</sub>, el grupo finito de
orden <code class="varname">q</code>, donde <code class="varname">q</code> es un primo. Si <code
class="varname">q</code> no es un primo los resultados son falsos.</p></dd><dt><span class="te
rm"><a
name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimitiveModWithPrimeFactors</span></dt><dd><pre
class="synopsis">IsPrimitiveModWithPrimeFactors (g,q,f)</pre><p>Comprobar si <code class="varname">g</code>
es primario en F<sub>q</sub>, el grupo finito de orden <code class="varname">q</code>, donde <code
class="varname">q</code> es un primo y <code class="varname">f</code> es un vector de factores primos de
<code class="varname">q</code>-1. Si <code class="varname">q</code> no es primo los resultados son
falsos.</p></dd><dt><span class="term"><a
name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre class="synopsis">IsPseudoprime
(n,b)</pre><p>Si <code class="varname">n</code> es pseudo-primo en base <code class="varname">b</code> pero
no un primo, esto es si <strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong>. Esto llama a <a
class="link" href="ch11s07.html#gel-function-PseudoprimeTest"><code class="function">PseudoprimeTest</co
de></a></p></dd><dt><span class="term"><a
name="gel-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Compruebe si <code class="varname">n</code> es un
pseudo-primo fuerte en base <code class="varname">b</code> pero no un primo.</p></dd><dt><span
class="term"><a name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi
(a,b)</pre><p>Alias: <code class="function">JacobiSymbol</code></p><p>Calcular el símbolo de Jacobi (a/b) (b
debe ser impar).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Alias: <code class="function">JacobiKroneckerSymbol</code></p><p>Calcular el símbolo de Jacobi
(a/b) con extensión de Kronecker (a/2)=(2/a) cuando sea impar, o (a/2)=0 cuando sea par.</p></dd><dt><span
class="term"><a name="gel-function-LeastAbsoluteResidue"></a>LeastAbsoluteResi
due</span></dt><dd><pre class="synopsis">LeastAbsoluteResidue (a,n)</pre><p>Devuelve el resto de <code
class="varname">a</code> mod <code class="varname">n</code> con el último valor absoluto (en el intervalo
-n/2 to n/2).</p></dd><dt><span class="term"><a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre
class="synopsis">Legendre (a,p)</pre><p>Alias: <code class="function">LegendreSymbol</code></p><p>Calcular el
símbolo de Legendre (a/p).</p><p>Consulte <a class="ulink" href="http://planetmath.org/LegendreSymbol";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Compruebe si 2<sup>p</sup>-1 es un primo de Mersenne utilizando la prueba de Lucas-Lehmer.
Consulte también <a class="link" href="ch11s07.html#gel
-function-MersennePrimeExponents">MersennePrimeExponents</a> y <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/LucasLhemer";
target="_top">Planetmath</a>, o <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Devuelve el <code class="varname">n</code>-ésimo número de Lucas.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfra
m.com/LucasNumber.html" target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span
class="term"><a name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Devuelve todos los factores primos de un
número.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>Un vector de Mersenne de exponentes primos conocidos, esto es
una lista de enteros positivos <code class="varname">p</code> tal que 2<sup>p</sup>-1 es un primo. Consulte
también <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>
y <a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>, <a
class="ulink" href="ht
tp://planetmath.org/MersenneNumbers" target="_top">Planetmath</a>, <a class="ulink"
href="http://mathworld.wolfram.com/MersennePrime.html"; target="_top">Mathworld</a> o <a class="ulink"
href="http://www.mersenne.org/"; target="_top">GIMPS</a> para obtener más información.</p></dd><dt><span
class="term"><a name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre
class="synopsis">MillerRabinTest (n,reps)</pre><p>Utiliza la prueba de números primarios Miller-Rabin de
<code class="varname">n</code>, <code class="varname">reps</code> número de veces. La probabilidad de falso
positivo es <strong class="userinput"><code>(1/4)^reps</code></strong>. Probablemente es mejor usar <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a> ya que es más
rápido y mejor sobre enteros más pequeños.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"; target="_top">Wikipedia
</a> o <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> o <a
class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>Utiliza la prueba Miller-Rabin de números primos de <code
class="varname">n</code> con las bases suficientes que asuman la hipótesis generalizada de Reimann, el
resultado es determinista.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>, o <a
class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> para obtener más in
formación.</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Devuelve el inverso de n módulo m.</p><p>Consulte <a class="ulink"
href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu
(n)</pre><p>Devuelve la función de Moebius «mu» de <code class="varname">n</code>. Esto es, devuelve 0 si
<code class="varname">n</code> no es un producto entre primos distintos y <strong
class="userinput"><code>(-1)^k</code></strong> si es un producto de <code class="varname">k</code> primos
distintos.</p><p>Consulte <a class="ulink" href="http://planetmath.org/MoebiusFunction";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html";
target="_top">Mathworld</a> para obten
er más información.</p></dd><dt><span class="term"><a
name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre class="synopsis">NextPrime
(n)</pre><p>Devuelve el primo menor más grande que <code class="varname">n</code>. Los primos negativos se
consideran primos y así para obtener el primo anterior, puede usar <strong
class="userinput"><code>-NextPrime(-n)</code></strong>.</p><p>Esta función utiliza las GMP <code
class="function">mpz_nextprime</code> la cual vuelve a utilizar la prueba probabilística de Miller-Rabin
(consulte también <a class="link" href="ch11s07.html#gel-function-MillerRabinTest"><code
class="function">MillerRabinTest</code></a>). La probabilidad de un falso positivo no se da, pero es lo
suficientemente baja para prácticamente todos los propósitos.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworl
d</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synopsis">PadicValuation
(n,p)</pre><p>Devuelve la evaluación del número «p-adic» (número de ceros que va dejando en base <code
class="varname">p</code>).</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/P-adic_order"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/PAdicValuation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre
class="synopsis">PowerMod (a,b,m)</pre><p>Calcula <strong class="userinput"><code>a^b mod m</code></strong>.
La potencia <code class="varname">b</code> de <code class="varname">a</code> módulo <code
class="varname">m</code>. No es necesario utilizar esta función ya que se utiliza automáticamente en modo
módulo. Por lo tanto <strong class
="userinput"><code>a^b mod m</code></strong> es igual de rápido.</p></dd><dt><span class="term"><a
name="gel-function-Prime"></a>Prime</span></dt><dd><pre class="synopsis">Prime (n)</pre><p>Alias: <code
class="function">prime</code></p><p>Devuelve el <code class="varname">n</code>-ésimo primo (hasta un
límite).</p><p>Consulte <a class="ulink" href="http://planetmath.org/PrimeNumber";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre class="synopsis">PrimeFactors
(n)</pre><p>Devuelve todos los factores primos de un número como un vector.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Prime_factor"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Mathworld</a> para obtener m
ás información.</p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">PseudoprimeTest
(n,b)</pre><p>Prueba de pseudo-primo, devuelve <code class="constant">true</code> sólo si <strong
class="userinput"><code>b^(n-1) == 1 mod n</code></strong></p><p>Consulte <a class="ulink"
href="http://planetmath.org/Pseudoprime"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/Pseudoprime.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre class="synopsis">RemoveFactor
(n,m)</pre><p>Elimina todas las instancias del factor <code class="varname">m</code> desde el número <code
class="varname">n</code>. Esto es, lo divide por la potencia mas grande de <code class="varname">m</code>,
que divide <code class="varname">n</code>.</p><p>Consulte <a class="ulink" href="http:
//planetmath.org/Divisibility" target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/Factor.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-SilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Buscar el logaritmo sencillo de
<code class="varname">n</code> base <code class="varname">b</code> en F<sub>q</sub>, de grupo de orden finito
<code class="varname">q</code>, donde <code class="varname">q</code> es un primo que utiliza el algoritmo de
Silver-Pohlig-Hellman, dado <code class="varname">f</code> es la factorización de <code
class="varname">q</code>-1.</p></dd><dt><span class="term"><a
name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Buscar la raíz cuadrada de <code class="varname">n</code> mó
dulo <code class="varname">p</code> (donde <code class="varname">p</code> es un primo). Se devuelve «null»
si el resto no es cuadrático.</p><p>Consulte <a class="ulink" href="http://planetmath.org/QuadraticResidue";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/QuadraticResidue.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Ejecutar la prueba del pseudo-primo fuerte en base <code
class="varname">b</code> de <code class="varname">n</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Strong_pseudoprime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/StrongPseudoprime"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/StrongPseudoprime.html"; target="_top">Mathwor
ld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-gcd"></a>gcd</span></dt><dd><pre class="synopsis">gcd (a,args...)</pre><p>Alias: <code
class="function">GCD</code></p><p>Máximo común divisor de enteros. Puede introducir tantos enteros en la
lista de argumentos, o puede introducir un vector o una matriz de enteros. Si introduce más de una matriz del
mismo tamaño, entonces el máximo común divisor se realiza elemento a elemento.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Greatest_common_divisor"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmath</a>, o <a
class="ulink" href="http://mathworld.wolfram.com/GreatestCommonDivisor.html"; target="_top">Mathworld</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-lcm"></a>lcm</span></dt><dd><pre class="synopsis">lcm (a,args...)</pre><p>Alias: <code
class
="function">LCM</code></p><p>Mínimo común múltiplo de enteros. Puede introducir tantos enteros en la lista
de argumentos, o introducir un vector o matriz de enteros. Si introduce mas de una matriz del mismo tamaño,
entonces el mínimo común múltiplo se realiza elemento a elemento.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Least_common_multiple"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LeastCommonMultiple"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/LeastCommonMultiple.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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Manipulación de matrices</td></tr></table></div></body></html>
+</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,tries)</pre><p>Probar la factorización de Fermat de <code
class="varname">n</code> en <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, devuelve <code
class="varname">t</code> y <code class="varname">s</code> como un vector si es posible, <code
class="constant">null</code> de otra manera <code class="varname">tries</code> especifica el número de
intentos antes de abandonar </p><p>Es una buena factorización si su número es el producto de dos factores que
están muy cerca.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Encontrar el primer elemento primitivo en F<sub>q</sub>,
en el grupo de orden finito<code class="varname">q</code>. Por supuesto, <code class="varname">q</code> debe
de ser primo.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Encontrar un elemento primitivo aleatorio en
F<sub>q</sub>, en el grupo de orden finito <code class="varname">q</code> (q debe de ser
primo)</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Calcula la base del logaritmo discreto <code class="varname">b</code> de n en
F<sub>q</sub>, el grupo finito de orden <code class="va
rname">q</code> (<code class="varname">q</code> un primo), utilizando el factor base <code
class="varname">S</code>. <code class="varname">S</code> será una columna de números primos y una segunda
columna precalculada por <a class="link" href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Ejecuta los pasos para los cálculos previos de
<a class="link" href="ch11s07.html#gel-function-IndexCalculus"><code
class="function">IndexCalculus</code></a> para logaritmos de base <code class="varname">b</code> en
F<sub>q</sub>, del grupo finito de orden <code class="varname">q</code> (<code class="varname">q</code> un
primo), para el factor base <code class="varname">S</code> (donde <code class="varname">S</code> es una co
lumna de vector de primos). Los registros se calculan previamente y se devuelven en la segunda
columna.</p></dd><dt><span class="term"><a name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre
class="synopsis">IsEven (n)</pre><p>Comprueba si un entero es par.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Comprueba si un entero positivo <code
class="varname">p</code> es un exponente primo de Mersenne. Esto es si 2<sup>p</sup>-1 es un primo. Esto lo
hace mirando en una tabla de valores conocidos que es relativamente corta. Vea también <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> y <a class="link"
href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
+ for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Comprueba si un número racional <code class="varname">m</code> es una potencia <code
class="varname">n</code>-ésima perfecta. Consulte <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> y <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Comprueba su un
entero es impar.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Comprobar si un entero es una potencia perfecta, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
+ Check an integer for being a perfect square of an integer. The number must
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
+ </p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>Comprueba si dos números enteros son primos, para números menores que
2.5e10 la respuesta es determinista (si la hipótesis de Riemann es verdadera). Para números más grandes, la
probabilidad de un falso positivo depende de <a class="link"
href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. Significa que la probabilidad de un falso positivo es 1/4
de la potencia <code class="function">IsPrimeMillerRabinReps</code>. De manera predeterminada el valor de 22
produce una probabilidad entorno a 5.7e-14.</p><p>Si se devuelve <code class="constant">false</code>, puede
estar seguro de que el número es un compuesto. Si quiere estar totalmente seguro de que tiene un número primo
use <a class="link" href="ch11s07.html#gel-function-MillerRabinTestSure"><code class="function">MillerRabinTe
stSure</code></a> pero esto le puede llevar mucho más tiempo.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre class="synopsis">IsPrimitiveMod
(g,q)</pre><p>Comprobar si <code class="varname">g</code> es primario en F<sub>q</sub>, el grupo finito de
orden <code class="varname">q</code>, donde <code class="varname">q</code> es un primo. Si <code
class="varname">q</code> no es un primo los resultados son falsos.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimitiveModWithPrimeFactors</span></dt><dd><pre
class="synopsis">IsPrimitiveModWithPrimeFactors (g,q,f)</pre><p>Comprobar si <code class="varname">g</code>
es primario en F<sub>q</sub>, el
grupo finito de orden <code class="varname">q</code>, donde <code class="varname">q</code> es un primo y
<code class="varname">f</code> es un vector de factores primos de <code class="varname">q</code>-1. Si <code
class="varname">q</code> no es primo los resultados son falsos.</p></dd><dt><span class="term"><a
name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre class="synopsis">IsPseudoprime
(n,b)</pre><p>Si <code class="varname">n</code> es pseudo-primo en base <code class="varname">b</code> pero
no un primo, esto es si <strong class="userinput"><code>b^(n-1) == 1 mod n</code></strong>. Esto llama a <a
class="link" href="ch11s07.html#gel-function-PseudoprimeTest"><code
class="function">PseudoprimeTest</code></a></p></dd><dt><span class="term"><a
name="gel-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Compruebe si <code class="varname">n</code> es un
pseudo-primo fuerte en base
<code class="varname">b</code> pero no un primo.</p></dd><dt><span class="term"><a
name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi (a,b)</pre><p>Alias: <code
class="function">JacobiSymbol</code></p><p>Calcular el símbolo de Jacobi (a/b) (b debe ser
impar).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Alias: <code class="function">JacobiKroneckerSymbol</code></p><p>Calcular el símbolo de Jacobi
(a/b) con extensión de Kronecker (a/2)=(2/a) cuando sea impar, o (a/2)=0 cuando sea par.</p></dd><dt><span
class="term"><a name="gel-function-LeastAbsoluteResidue"></a>LeastAbsoluteResidue</span></dt><dd><pre
class="synopsis">LeastAbsoluteResidue (a,n)</pre><p>Devuelve el resto de <code class="varname">a</code> mod
<code class="varname">n</code> con el último valor absoluto (en el intervalo -n/2 to n/2).</p></dd><dt><span
class="term"
<a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre class="synopsis">Legendre
(a,p)</pre><p>Alias: <code class="function">LegendreSymbol</code></p><p>Calcular el símbolo de Legendre
(a/p).</p><p>Consulte <a class="ulink" href="http://planetmath.org/LegendreSymbol";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Compruebe si 2<sup>p</sup>-1 es un primo de Mersenne utilizando la prueba de Lucas-Lehmer.
Consulte también <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> y <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Devuelve el <code class="varname">n</code>-ésimo número de Lucas.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Devuelve todos los factores primos de un
número.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>Un vector de Mersenne de exponentes primos conocidos, esto es
una lista de enteros positivos <code class="varname">p</code> tal que 2<sup>p</sup>-1 es un primo. Consulte
también <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>
y <a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
+ for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre class="synopsis">MillerRabinTest
(n,reps)</pre><p>Utiliza la prueba de números primarios Miller-Rabin de <code class="varname">n</code>, <code
class="varname">reps</code> número de veces. La probabilidad de falso positivo es <strong
class="userinput"><code>(1/4)^reps</code></strong>. Probablemente es mejor usar <a class="link"
href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a> ya que es más rápido y
mejor sobre enteros más pequeños.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>Utiliza la prueba Miller-Rabin de números primos de <code
class="varname">n</code> con las bases suficientes que asuman la hipótesis generalizada de Reimann, el
resultado es determinista.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
+ <a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Devuelve el inverso de n módulo m.</p><p>Consulte <a class="ulink"
href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu
(n)</pre><p>Devuelve la función de Moebius «mu» de <code class="varname">n</code>. Esto es, devuelve 0 si
<code class="varname">n</code> no es un producto entre primos distintos y <strong
class="userinput"><code>(-1)^k</code></strong> si es un producto de <code class="varname">k</code> primos
distintos.</p><p>Consulte <a class="ulink" href="http://planetmath.org/MoebiusFunction";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html";
target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre class="synopsis">NextPrime
(n)</pre><p>Devuelve el primo menor más grande que <code class="varname">n</code>. Los primos negativos se
consideran primos y así para obtener el primo anterior, puede usar <strong
class="userinput"><code>-NextPrime(-n)</code></strong>.</p><p>Esta función utiliza las GMP <code
class="function">mpz_nextprime</code> la cual vuelve a utilizar la prueba probabilística de Miller-Rabin
(consulte también <a class="link" href="ch11s07.html#gel-function-MillerRabinTest"><code
class="function">MillerRabinTest</code></a>). La probabilidad de un falso positivo no se da, pero es lo
suficientemente baja para prácticamente todos los propósitos.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> pa
ra obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synopsis">PadicValuation
(n,p)</pre><p>Devuelve la evaluación del número «p-adic» (número de ceros que va dejando en base <code
class="varname">p</code>).</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/P-adic_order"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/PAdicValuation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre
class="synopsis">PowerMod (a,b,m)</pre><p>Calcula <strong class="userinput"><code>a^b mod m</code></strong>.
La potencia <code class="varname">b</code> de <code class="varname">a</code> módulo <code
class="varname">m</code>. No es necesario utilizar esta función ya que se utiliza automáticamente en modo
módulo. Por lo tanto <strong class="userin
put"><code>a^b mod m</code></strong> es igual de rápido.</p></dd><dt><span class="term"><a
name="gel-function-Prime"></a>Prime</span></dt><dd><pre class="synopsis">Prime (n)</pre><p>Alias: <code
class="function">prime</code></p><p>Devuelve el <code class="varname">n</code>-ésimo primo (hasta un
límite).</p><p>Consulte <a class="ulink" href="http://planetmath.org/PrimeNumber";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre class="synopsis">PrimeFactors
(n)</pre><p>Devuelve todos los factores primos de un número como un vector.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Prime_factor"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Mathworld</a> para obtener
más info
rmación.</p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">PseudoprimeTest
(n,b)</pre><p>Prueba de pseudo-primo, devuelve <code class="constant">true</code> sólo si <strong
class="userinput"><code>b^(n-1) == 1 mod n</code></strong></p><p>Consulte <a class="ulink"
href="http://planetmath.org/Pseudoprime"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/Pseudoprime.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre class="synopsis">RemoveFactor
(n,m)</pre><p>Elimina todas las instancias del factor <code class="varname">m</code> desde el número <code
class="varname">n</code>. Esto es, lo divide por la potencia mas grande de <code class="varname">m</code>,
que divide <code class="varname">n</code>.</p><p>Consulte <a class="ulink" href="http://planet
math.org/Divisibility" target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/Factor.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-SilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Buscar el logaritmo sencillo de
<code class="varname">n</code> base <code class="varname">b</code> en F<sub>q</sub>, de grupo de orden finito
<code class="varname">q</code>, donde <code class="varname">q</code> es un primo que utiliza el algoritmo de
Silver-Pohlig-Hellman, dado <code class="varname">f</code> es la factorización de <code
class="varname">q</code>-1.</p></dd><dt><span class="term"><a
name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Buscar la raíz cuadrada de <code class="varname">n</code> módulo <co
de class="varname">p</code> (donde <code class="varname">p</code> es un primo). Se devuelve «null» si el
resto no es cuadrático.</p><p>Consulte <a class="ulink" href="http://planetmath.org/QuadraticResidue";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/QuadraticResidue.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Ejecutar la prueba del pseudo-primo fuerte en base <code
class="varname">b</code> de <code class="varname">n</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Strong_pseudoprime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/StrongPseudoprime"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/StrongPseudoprime.html"; target="_top">Mathworld</a> p
ara obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-gcd"></a>gcd</span></dt><dd><pre class="synopsis">gcd (a,args...)</pre><p>Alias: <code
class="function">GCD</code></p><p>Máximo común divisor de enteros. Puede introducir tantos enteros en la
lista de argumentos, o puede introducir un vector o una matriz de enteros. Si introduce más de una matriz del
mismo tamaño, entonces el máximo común divisor se realiza elemento a elemento.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Greatest_common_divisor"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmath</a>, o <a
class="ulink" href="http://mathworld.wolfram.com/GreatestCommonDivisor.html"; target="_top">Mathworld</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-lcm"></a>lcm</span></dt><dd><pre class="synopsis">lcm (a,args...)</pre><p>Alias: <code
class="functi
on">LCM</code></p><p>Mínimo común múltiplo de enteros. Puede introducir tantos enteros en la lista de
argumentos, o introducir un vector o matriz de enteros. Si introduce mas de una matriz del mismo tamaño,
entonces el mínimo común múltiplo se realiza elemento a elemento.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Least_common_multiple"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LeastCommonMultiple"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/LeastCommonMultiple.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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name="genius-gel-function-list-matrix"></a>Manipulación de matrices</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Aplicar una función sobre todos los elementos de una matriz y devolver una matriz con los
resultados.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Aplicar una función sobre todos los elementos de dos
matrices (o un valor y una matriz) y devolver una matriz con los resultados.</p></dd><dt><span
class="term"><a name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf
(M)</pre><p>Obtener las columnas de una matriz como un vector horizontal.</p></dd><dt><span class="term"><a
nam
e="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre
class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Eliminar filas y columnas de una
matriz.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Calcular la k-ésima matriz compuesta de A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>Contar el número de cero columnas en una matriz. Por ejemplo
una vez que su columna reduce una matriz puede usar esto para encontrar la nulidad. Consulte <a class="link"
href="ch11s09.html#gel-function-cref"><code class="function">cref</code></a> y <a class="link"
href="ch11s09.html#gel-function-Nullity"><code class="function">Nullity</code></a>.</p></dd><dt><span
class="term"><a name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre cla
ss="synopsis">DeleteColumn (M,col)</pre><p>Eliminar una columna de una matriz.</p></dd><dt><span
class="term"><a name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Eliminar una fila de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Obtener las entradas diagonales de una matriz como un vector columna.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices"; target="_top">Wikipedia</a>
para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Obtener el producto escalar de dos vectores. Los vectores serán del mismo tamaño. Se toman no
conjugados por lo que tendrá forma bilineal incluso si se trabaja con números complejos. Esto es el producto
escalar bilineal
, no el producto escalar sesquilienal. Consulte <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> para el producto interno estándar
sesquilinear.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Dot_product";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/DotProduct";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre class="synopsis">ExpandMatrix
(M)</pre><p>Expandir una matriz de la misma manera que hacemos con la entrada sin comillas de la matriz. Esto
es, se expande cualquier matriz interna como bloques. Esto es una manera de construir matrices fuera de las
mas pequeñas y se hace de manera automática en la entrada a menos que la matriz se
entrecomille.</p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre class="syn
opsis">HermitianProduct (u,v)</pre><p>Alias: <code class="function">InnerProduct</code></p><p>Obtener el
producto de Hermitian de dos vectores. Los vectores serán del mismo tamaño. Esto es una forma «sesquilinear»
para utilizar la identidad de la matriz.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Sesquilinear_form"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/HermitianInnerProduct.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-I"></a>I</span></dt><dd><pre
class="synopsis">I (n)</pre><p>Alias: <code class="function">eye</code></p><p>Devolver una matriz identidad
del tamaño dado, es decir, de <code class="varname">n</code> por <code class="varname">n</code>. Si <code
class="varname">n</code> es cero, devuelve <code class="constant">null</code>.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Identity_matrix"; target
="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/IdentityMatrix";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vec,msize)</pre><p>Devuelve el complemento índice de un vector de índices. Todo en base a uno. Por ejemplo
para el vector <strong class="userinput"><code>[2,3]</code></strong> y tamaño <strong
class="userinput"><code>5</code></strong>, devolverá <strong class="userinput"><code>[1,4,5]</code></strong>.
Si <code class="varname">msize</code> es 0, siempre devolverá <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Es
una matriz diagonal.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix";
target="_top">Wikipedia</a> o <a class="ul
ink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Comprobar si una matriz es la matriz de identidad. Automáticamente devuelve <code
class="constant">false</code> si la matriz no es cuadrada. También trabaja con números, en cualquier caso
este es equivalente a <strong class="userinput"><code>x==1</code></strong>. Cuando <code
class="varname">x</code> es <code class="constant">null</code> (imaginemos que es como una matriz de 0 por
0), no se genera error y se devuelve <code class="constant">false</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Es una matriz triangular inferior. Esto es, todas las entradas
están por encima de la diagonal cero.</p></dd><dt><sp
an class="term"><a name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre
class="synopsis">IsMatrixInteger (M)</pre><p>Comprobar si una matriz es una matriz de enteros (no
compleja).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Comprobar si una matriz no es negativa, es decir, si cada
elemento no es negativo. No confunda matrices positivas con matrices semidefinidas positivas.</p><p>Consulte
la <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Comprobar si una matriz es positiva, es decir, si cada elemento
es positivo (y por lo tanto real). Individualmente, ningún elemento es 0. No confunda matrices p
ositivas con matrices definidas positivas.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Comprobar si el argumento es una matriz de números racionales
(no complejos)</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Comprobar si el argumento es una matriz de números reales (no complejos).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Comprobar si una matriz es cuadrada, es decir, si su altura es
igual a su anchura.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><d
d><pre class="synopsis">IsUpperTriangular (M)</pre><p>¿Es una matriz triangular superior?. Esto se cumple si
todas las entradas por debajo de la diagonal son cero.</p></dd><dt><span class="term"><a
name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Comprobar si una matriz es una matriz de sólo números. Muchas funciones internas hacen esta
comprobación. Los valores pueden ser cualquier número, incluyendo números complejos.</p></dd><dt><span
class="term"><a name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector
(v)</pre><p>Indica si el argumento de un vector es horizontal o vertical. Genius no distingue entre una
matriz y un vector, y un vector es justo una matriz 1 por <code class="varname">n</code> o <code
class="varname">n</code> por 1.</p></dd><dt><span class="term"><a
name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero (x)</pre><p>Comprobar si una
matriz está compuesta toda por ceros. También trabaja con números, en cualquier caso esto es equivalente a
<strong class="userinput"><code>x==0</code></strong>. Cuando <code class="varname">x</code> es <code
class="constant">null</code> (imagine que es una matriz de 0 por 0), no se genera ningún error y devuelve
<code class="constant">true</code> que indica que la matriz está compuesta de ceros.</p></dd><dt><span
class="term"><a name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre
class="synopsis">LowerTriangular (M)</pre><p>Devuelve una copia de la matriz <code class="varname">M</code>
con todas las entradas por encima de la diagonal establecidas a cero.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Alias: <code class="function">diag</code></p><p>Hacer una matriz diagonal desde un vector.
Alternativamente puede pasarle los valores como argu
mentos para la diagonal. Así <strong class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> es lo
mismo que <strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Hacer un vector columna fuera de la matriz colocando columnas una encima de la otra. Devuelve
<code class="constant">null</code> cuando se introduce <code class="constant">null</code>.</p></dd><dt><span
class="term"><a name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre
class="synopsis">MatrixProduct (A)</pre><p>Calcular el producto de todos los elementos en una matriz o
vector. Es decir, multiplicar t
odos los elementos y devolver un número que es el producto de todos los elementos.</p></dd><dt><span
class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre class="synopsis">MatrixSum
(A)</pre><p>Calcular la suma de todos los elementos en una matriz o vector. Es decir, sumar todos los
elementos y devolver un número que es el resultado de la suma de todos los elementos.</p></dd><dt><span
class="term"><a name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Calcular la suma de los cuadrados de todos los elementos en una
matriz o vector.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Devuelve una fila vector de índices de columnas distintas de cero en la matriz <code
class="varname">M</code>.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-
function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Devuelve una fila vector de índices de elementos distintos de cero en el vector <code
class="varname">v</code>.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Obtener el producto externo de dos vectores. Esto es, suponga que <code
class="varname">u</code> y <code class="varname">v</code> son vectores verticales, entonces el producto
externo es <strong class="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Invierte el orden de los elementos de un vector (devuelve <code class="constant">null</code> si
se le pasa <code class="constant">null</code>).</p></dd><dt><span class="term"><a n
ame="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Calcula la suma
de cada fila de una matriz y devuelve el resultado en un vector vertical con el resultado</p></dd><dt><span
class="term"><a name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre
class="synopsis">RowSumSquares (m)</pre><p>Calcular la suma de los cuadrados de cada fila de una matriz y
devolver una matriz columna con los resultados.</p></dd><dt><span class="term"><a
name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre class="synopsis">RowsOf (M)</pre><p>Obtiene las
filas de una matriz como un vector vertical. Cada elemento del vector es un vector horizontal que se
corresponde con la fila de <code class="varname">M</code>. Esta función es útil si se quiere recorrer las
filas de una matriz. Por ejemplo, como en <strong class="userinput"><code>for r in RowsOf(M) do
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align="left"><a accesskey="p" href="ch11s07.html">Anterior</a> </td><th width="60%" align="center">Capítulo
11. Lista de funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
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page"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Manipulación de matrices</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Aplicar una función sobre todos los elementos de una matriz y devolver una matriz con los
resultados.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Aplicar una función sobre todos los elementos de dos
matrices (o un valor y una matriz) y devolver una matriz con los resultados.</p></dd><dt><span
class="term"><a name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf
(M)</pre><p>Obtener las columnas de una matriz como un vector horizontal.</p></dd><dt><span class="term"><a
nam
e="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre
class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Eliminar filas y columnas de una
matriz.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Calcular la k-ésima matriz compuesta de A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
+ the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
+ and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Eliminar una columna de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Eliminar una fila de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Obtener las entradas diagonales de una matriz como un vector columna.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Obtener el producto escalar de dos vectores. Los vectores serán del mismo tamaño. Se toman no
conjugados por lo que tendrá forma bilineal incluso si se trabaja con números complejos. Esto es el producto
escalar bilineal, no el producto escalar sesquilienal. Consulte <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> para el producto interno estándar
sesquilinear.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Dot_product";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/DotProduct";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre class="synopsis">ExpandMatrix
(M)</pre><p>Expandir una matriz de la misma manera que hacemos
con la entrada sin comillas de la matriz. Esto es, se expande cualquier matriz interna como bloques. Esto es
una manera de construir matrices fuera de las mas pequeñas y se hace de manera automática en la entrada a
menos que la matriz se entrecomille.</p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre
class="synopsis">HermitianProduct (u,v)</pre><p>Alias: <code
class="function">InnerProduct</code></p><p>Obtener el producto de Hermitian de dos vectores. Los vectores
serán del mismo tamaño. Esto es una forma «sesquilinear» para utilizar la identidad de la
matriz.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Sesquilinear_form";
target="_top">Wikipedia</a> o <a class="ulink" href="http://mathworld.wolfram.com/HermitianInnerProduct.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-I"></a>I</span></dt><dd><pre class="synop
sis">I (n)</pre><p>Alias: <code class="function">eye</code></p><p>Devolver una matriz identidad del tamaño
dado, es decir, de <code class="varname">n</code> por <code class="varname">n</code>. Si <code
class="varname">n</code> es cero, devuelve <code class="constant">null</code>.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Identity_matrix"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vec,msize)</pre><p>Devuelve el complemento índice de un vector de índices. Todo en base a uno. Por ejemplo
para el vector <strong class="userinput"><code>[2,3]</code></strong> y tamaño <strong
class="userinput"><code>5</code></strong>, devolverá <strong class="userinput"><code>[1,4,5]</code></strong>.
Si <code class="
varname">msize</code> es 0, siempre devolverá <code class="constant">null</code>.</p></dd><dt><span
class="term"><a name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal
(M)</pre><p>Es una matriz diagonal.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Comprobar si una matriz es la matriz de identidad. Automáticamente devuelve <code
class="constant">false</code> si la matriz no es cuadrada. También trabaja con números, en cualquier caso
este es equivalente a <strong class="userinput"><code>x==1</code></strong>. Cuando <code
class="varname">x</code> es <code class="constant">null</code> (imaginemos que es como una matriz de 0 por
0), no se genera error y se devuelve <code class="constant">false</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Es una matriz triangular inferior. Esto es, todas las entradas
están por encima de la diagonal cero.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="syno
psis">IsMatrixInteger (M)</pre><p>Comprobar si una matriz es una matriz de enteros (no
compleja).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Comprobar si una matriz no es negativa, es decir, si cada
elemento no es negativo. No confunda matrices positivas con matrices semidefinidas positivas.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Comprobar si una matriz es positiva, es decir, si cada elemento
es positivo (y por lo tanto real). Individualmente, ningún elemento es 0. No confunda matrices positivas con
matrices definidas positivas.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Comprobar si el argumento es una matriz de números racionales
(no complejos)</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Comprobar si el argumento es una matriz de números reales (no complejos).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Comprobar si una matriz es cuadrada, es decir, si su altura es
igual a su anchura.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>¿Es una matriz triangular superior?. Esto se cumple si todas
las entradas por debajo de la diagonal son cero.</p></dd><dt><span cla
ss="term"><a name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre
class="synopsis">IsValueOnly (M)</pre><p>Comprobar si una matriz es una matriz de sólo números. Muchas
funciones internas hacen esta comprobación. Los valores pueden ser cualquier número, incluyendo números
complejos.</p></dd><dt><span class="term"><a name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre
class="synopsis">IsVector (v)</pre><p>Indica si el argumento de un vector es horizontal o vertical. Genius no
distingue entre una matriz y un vector, y un vector es justo una matriz 1 por <code class="varname">n</code>
o <code class="varname">n</code> por 1.</p></dd><dt><span class="term"><a
name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero (x)</pre><p>Comprobar si
una matriz está compuesta toda por ceros. También trabaja con números, en cualquier caso esto es equivalente
a <strong class="userinput"><code>x==0</code></strong>. Cuando <code class="v
arname">x</code> es <code class="constant">null</code> (imagine que es una matriz de 0 por 0), no se genera
ningún error y devuelve <code class="constant">true</code> que indica que la matriz está compuesta de
ceros.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Devuelve una copia de la matriz <code class="varname">M</code> con todas las entradas por encima
de la diagonal establecidas a cero.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Alias: <code class="function">diag</code></p><p>Hacer una matriz diagonal desde un vector.
Alternativamente puede pasarle los valores como argumentos para la diagonal. Así <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> es lo mismo que <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</
p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Hacer un vector columna fuera de la matriz colocando columnas una encima de la otra. Devuelve
<code class="constant">null</code> cuando se introduce <code class="constant">null</code>.</p></dd><dt><span
class="term"><a name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre
class="synopsis">MatrixProduct (A)</pre><p>Calcular el producto de todos los elementos en una matriz o
vector. Es decir, multiplicar todos los elementos y devolver un número que es el producto de todos los
elementos.</p></dd><dt><span class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre
class="synopsis">MatrixSum (A)</pre><p>Calcular la suma de todos los elementos en una matriz o vector. Es
decir, sumar todos los elementos y devolver un número que es el resultado de la suma de todos los
elementos.</p></dd><dt><sp
an class="term"><a name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Calcular la suma de los cuadrados de todos los elementos en una
matriz o vector.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Devuelve una fila vector de índices de columnas distintas de cero en la matriz <code
class="varname">M</code>.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Devuelve una fila vector de índices de elementos distintos de cero en el vector <code
class="varname">v</code>.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre>
<p>Obtener el producto externo de dos vectores. Esto es, suponga que <code class="varname">u</code> y <code
class="varname">v</code> son vectores verticales, entonces el producto externo es <strong
class="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Invierte el orden de los elementos de un vector (devuelve <code class="constant">null</code> si
se le pasa <code class="constant">null</code>).</p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Calcula la suma
de cada fila de una matriz y devuelve el resultado en un vector vertical con el resultado</p></dd><dt><span
class="term"><a name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre
class="synopsis">RowSumSquares (m)</pre><p>Calcular la suma de los cuadrados de cada fila de una matriz y de
volver una matriz columna con los resultados.</p></dd><dt><span class="term"><a
name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre class="synopsis">RowsOf (M)</pre><p>Obtiene las
filas de una matriz como un vector vertical. Cada elemento del vector es un vector horizontal que se
corresponde con la fila de <code class="varname">M</code>. Esta función es útil si se quiere recorrer las
filas de una matriz. Por ejemplo, como en <strong class="userinput"><code>for r in RowsOf(M) do
something(r)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-SetMatrixSize"></a>SetMatrixSize</span></dt><dd><pre class="synopsis">SetMatrixSize
(M,filas,columnas)</pre><p>Hacer una nueva matriz del mismo tamaño que otra. Es decir, devolverá una nueva
matriz con la copia de otra. Las entradas que no caben, se recortan y el espacio adicional se rellena con
ceros. Si <code class="varname">rows</code> o <code class="varname">columns</code> son cero, entonces se
devuelve<code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-ShuffleVector"></a>ShuffleVector</span></dt><dd><pre class="synopsis">ShuffleVector
(v)</pre><p>Mezcla los elementos en un vector. Devuelve <code class="constant">null</code> si se le pasa
<code class="constant">null</code>.</p><p>Desde la versión 1.0.13 en adelante.</p></dd><dt><span
class="term"><a name="gel-function-SortVector"></a>SortVector</span></dt><dd><pre class="synopsis">SortVector
(v)</pre
<p>Ordenar los elementos del vector en orden ascendente.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroColumns"></a>StripZeroColumns</span></dt><dd><pre
class="synopsis">StripZeroColumns (M)</pre><p>Quita todas las columnas de ceros de <code
class="varname">M</code>.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroRows"></a>StripZeroRows</span></dt><dd><pre class="synopsis">StripZeroRows
(M)</pre><p>Quita todas las filas de ceros de <code class="varname">M</code>.</p></dd><dt><span
class="term"><a name="gel-function-Submatrix"></a>Submatrix</span></dt><dd><pre class="synopsis">Submatrix
(m,r,c)</pre><p>Devolver columnas y filas desde una matriz. Esto es equivalente a <strong
class="userinput"><code>m@(r,c)</code></strong>. <code class="varname">r</code> y <code
class="varname">c</code> serán vectores de filas y columnas (o números sencillos si sólo se necesita una
fila o columna).</p></dd><dt><span class="term"><a name="gel-function-SwapRows
"></a>SwapRows</span></dt><dd><pre class="synopsis">SwapRows (m,fila1,fila2)</pre><p>Intercambiar dos
columnas de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-UpperTriangular"></a>UpperTriangular</span></dt><dd><pre class="synopsis">UpperTriangular
(M)</pre><p>Devuelve una copia de la matriz <code class="varname">M</code> con todas las entradas por debajo
de la diagonal establecidas a cero.</p></dd><dt><span class="term"><a
name="gel-function-columns"></a>columns</span></dt><dd><pre class="synopsis">columns (M)</pre><p>Obtener el
número de columnas de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-elements"></a>elements</span></dt><dd><pre class="synopsis">elements (M)</pre><p>Obtener
el número total de elementos de una matriz. Es decir, el número de columnas por el número de
filas.</p></dd><dt><span class="term"><a name="gel-function-ones"></a>ones</span></dt><dd><pre
class="synopsis">ones (filas,columnas...)</pre><p>Hacer una matr
iz rellena de unos (o un vector fila si sólo se introduce un argumento). Devuelve <code
class="constant">null</code> si cualquier fila o columna es cero.</p></dd><dt><span class="term"><a
name="gel-function-rows"></a>rows</span></dt><dd><pre class="synopsis">rows (M)</pre><p>Obtener el número de
filas de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-zeros"></a>zeros</span></dt><dd><pre class="synopsis">zeros
(filas,columnas...)</pre><p>Hacer una matriz llena de ceros (o un vector fila si se introduce sólo un
argumento). Devuelve <code class="constant">null</code> si cualquier fila o columna es
cero.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-function-list-linear-algebra"></a>Álgebra
lineal</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Obtener la matriz auxiliar de tamaño <code
class="varname">n</code>. Esto es una matriz cuadrada que es toda ceros excepto la superdiagonal, que son
todos unos. Es la matriz de bloques de Jordan de un cero como valor propio.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> para obtener más información
sobre la forma canónica de Jordan.</p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm (v,A,w)
</pre><p>Evaluar (v,w) con respecto a la forma bilineal dada por la matriz A.</p></dd><dt><span
class="term"><a name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Devolver una función que evalúa dos vectores con respecto
a la forma bilineal dada por A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Alias: <code
class="function">CharPoly</code></p><p>Obtener el polinomio característico como un vector. Es decir, devuelve
los coeficientes del polinomio empezando por el término constante. Este polinomio se define por <strong
class="userinput"><code>det(M-xI)</code></strong>. Las raíces de este polinomio tienen como valor propio a
<code class="varname">M</code>. Consulte <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">Chara
cteristicPolynomialFunction</a>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Obtener el polinomio característico como una
función. Es decir, el polinomio se define por <strong class="userinput"><code>det(M-xI)</code></strong>. Las
raíces de este polinomio tienen un valor propio de <code class="varname">M</code>. Consulte <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> o <a
class="u
link" href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre class="synopsis">ColumnSpace
(M)</pre><p>Obtener una matriz base para el espacio de la columna de una matriz. Es decir, devuelve una
matriz la cual las columnas son las bases para el espacio de la columna <code class="varname">M</code>. Esto
es el espacio generado por las columnas de <code class="varname">M</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre
class="synopsis">CommutationMatrix (m, n)</pre><p>Devolver la matriz de conmutación <strong
class="userinput"><code>K(m,n)</code></strong> que es la única matriz <strong class="us
erinput"><code>m*n</code></strong> por <strong class="userinput"><code>m*n</code></strong> tal que <strong
class="userinput"><code>K(m,n) * MakeVector(A) = MakeVector(A.')</code></strong> para todas las matrices
<code class="varname">A</code> <code class="varname">m</code> por <code
class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre class="synopsis">CompanionMatrix
(p)</pre><p>Matriz acompañante de un polinomio (como vector).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Conjugada traspuesta de una matriz (adjunta). Es lo mismo que
el operador <strong class="userinput"><code>'</code></strong>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Conjugate_transpose"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/ConjugateTranspo
se" target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Alias: <code class="function">convol</code></p><p>Calcular la convolución de dos vectores
horizontales.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Calcular la convolución de dos vectores horizontales.
Devuelve el resultado como un vector y no se suman.</p></dd><dt><span class="term"><a
name="gel-function-CrossProduct"></a>CrossProduct</span></dt><dd><pre class="synopsis">CrossProduct
(v,w)</pre><p>Producto cruzado de dos vectores en R<sup>3</sup> como un vector columna.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Cross_product"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a n
ame="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre
class="synopsis">DeterminantalDivisorsInteger (M)</pre><p>Obtiene determinantes divisores de una matriz de
enteros.</p></dd><dt><span class="term"><a name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre
class="synopsis">DirectSum (M,N...)</pre><p>Suma directa de matrices.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-DirectSumMatrixVector"></a>DirectSumMatrixVector</span></dt><dd><pre
class="synopsis">DirectSumMatrixVector (v)</pre><p>Suma directa de un vector de matrices.</p><p>Consulte la
<a class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a>
para obtener más información.</p></dd><dt><span class="term"><a name="gel-function-Eigenvalues"></a>Eigenv
alues</span></dt><dd><pre class="synopsis">Eigenvalues (M)</pre><p>Alias: <code
class="function">eig</code></p><p>Obtener los valores propios de una matriz cuadrada. En la actualidad solo
funciona con matrices de tamaño 4 por 4 como máximo, o para matrices triangulares (cuyo valores propios están
en la diagonal).</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/Eigenvalue";
target="_top">Planetmath</a>, o <a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Obtener los autovectores de una matriz cuadrada. Op
cionalmente, obtener los autovalores y su multiplicidad algebraica. Actualmente funciona sólo para matrices
de hasta 2x2.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/Eigenvector";
target="_top">Planetmath</a>, o <a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Aplicar el proceso de Gram-Schmidt (a las columnas) con respecto al propio producto dado por
<code class="varname">B</code>. Si <code class="varname">B</code> no se da, entonces se utiliza el producto
Hermitiano estándar. <code class="varname">B</code> también puede ser una función sesquilineal de dos
argumentos o puede ser una matriz que devuelve una forma sesquilineal. Los vecto
res serán ortonormales con respecto a <code class="varname">B</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/GramSchmidtOrthogonalization"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span></dt><dd><pre class="synopsis">HankelMatrix
(c,r)</pre><p>La matriz de Hankel es una matriz cuyas diagonales (de izquierda a derecha) son constantes. La
primera fila es <code class="varname">c</code> y la última colúmna es <code class="varname">r</code>. Se
considera que ambos argumentos son vectores y que el último elemento de la fila <code
class="varname">c</code> es el mismo que el primer elemento de la columna <code
class="varname">r</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hankel_matrix"; target="_top">Wikiped
ia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Matriz de Hilbert de orden <code class="varname">n</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-Image"></a>Image</span></dt><dd><pre
class="synopsis">Image (T)</pre><p>Obtener la imagen (espacio columna) de una transformación
lineal.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Row_and_column_spaces";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre class="synopsis">InfNorm (v)</pre><p>Obtener el
operador norma d
e un vector, a veces también se denomina norma suprema o norma máxima.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Obtiene los factores invariantes de una matriz cuadrada
de enteros.</p></dd><dt><span class="term"><a
name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatrix</span></dt><dd><pre
class="synopsis">InverseHilbertMatrix (n)</pre><p>Matriz inversa de Hilbert de orden <code
class="varname">n</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Es una matriz Hermitian. Es decir, es igual a su tr
aspuesta conjugada.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Hermitian_matrix";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/HermitianMatrix";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IsInSubspace"></a>IsInSubspace</span></dt><dd><pre class="synopsis">IsInSubspace
(v,W)</pre><p>Comprueba si un vector está en un subespacio.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Es una matriz (o número) invertible (La matriz de enteros es invertible si, y sólo si esta es
invertible sobre los enteros).</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Es una matriz (o un número) inversible sobre un
campo.</p></dd><dt><span class="term"><a name="g
el-function-IsNormal"></a>IsNormal</span></dt><dd><pre class="synopsis">IsNormal (M)</pre><p>Indica que
<code class="varname">M</code> es una matriz normal. Es decir, realiza <strong class="userinput"><code>M*M'
== M'*M</code></strong>.</p><p>Consulte <a class="ulink" href="http://planetmath.org/NormalMatrix";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/NormalMatrix.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Indica que <code class="varname">M</code> es una matriz
definida positiva Hermitiana. Esto es si <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> es siempre estrictamente positivo para
cualquier vector <code class="varname">v</code>. <code class="varname">M</code> será cuadrada y Hermitiana
para ser definida positiva. La compr
obación de que se lleva a cabo es que cada submatriz principal tiene un determinante no negativo. (Consulte
<a class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Tenga en cuenta
que algunos autores (por ejemplo Mathworld) no requieren que <code class="varname">M</code> sea Hermitiana, y
entonces la condición está en la parte real del propio producto, pero aquí no se compartirá este punto de
vista. Si quiere comprobarlo, hacer sólo la parte Hermitiana de la matriz <code class="varname">M</code> como
sigue: <strong class="userinput"><code>IsPositiveDefinite(M+M')</code></strong>.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Positive-definite_matrix"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html"; target="_top">Mathworld</a> para obtener más
informaci�
�n.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Indica si <code class="varname">M</code> es una matriz
semidefinida positiva Hermitiana. Esto es si <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> es siempre no negativo para cualquier vector
<code class="varname">v</code>. <code class="varname">M</code> será cuadrada y Hermitiana para ser
semidefinida positiva. La comprobación que se lleva a cabo es que cada submatriz principal tenga un
determinante no negativo. (Consulte <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Tenga en cuenta que algunos
autores no requieren que <code class="varname">M</code> sea Hermitiana, y entonces la condición está en la
parte real del propio producto, pero aquí no se compartirá este punto de vista. Si quiere comprobarlo, hacer
sólo la part
e Hermitiana de la matriz <code class="varname">M</code> como sigue: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html"; target="_top">Mathworld</a> para obtener
más información.</p></dd><dt><span class="term"><a
name="gel-function-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian
(M)</pre><p>Es matriz antihermítica. Esto es, la transposición conjugada es igual al negativo de la
matriz.</p><p>Consulte <a class="ulink" href="http://planetmath.org/SkewHermitianMatrix";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre class="synopsis">IsUnitary (M)</pre><p>¿Es
una matriz unitaria?. Esto es, hacer <strong cl
ass="userinput"><code>M'*M</code></strong> y <strong class="userinput"><code>M*M'</code></strong> igual a la
identidad.</p><p>Consulte <a class="ulink" href="http://planetmath.org/UnitaryTransformation";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/UnitaryMatrix.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-JordanBlock"></a>JordanBlock</span></dt><dd><pre class="synopsis">JordanBlock
(n,lambda)</pre><p>Alias: <code class="function">J</code></p><p>Obtener el bloque de Jordan correspondiente
al valor propio <code class="varname">lambda</code> con multiplicidad <code
class="varname">n</code>.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><
a name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre class="synopsis">Kernel (T)</pre><p>Obtener el
núcleo (espacio nulo) de una trasformación lineal.</p><p>(Consulte <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Alias: <code
class="function">TensorProduct</code></p><p>Calcula el producto de Kronecker (producto tensorial en base
estándar) de dos matrices.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Kronecker_product"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/KroneckerProduct.html"; target="_top">Mathworld</a> para obtener más
información.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a name=
"gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition (A,
L, U)</pre><p>Obtener la descomposición de LU de <code class="varname">A</code> es decir, encontrar una
matriz triangular inferior y la matriz triangular superior cuyo producto es <code class="varname">A</code>.
Guarda el resultado en <code class="varname">L</code> y <code class="varname">U</code> que son referencias.
Devuelve <code class="constant">true</code> si se completó con éxito. Por ejemplo, suponga que «A» es una
matriz cuadrada, entonces después ejecute: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>LUDecomposition(A,&L,&U)</code></strong>
-</pre><p> tendrá la matriz inferior guardada en una variable llamada <code class="varname">L</code> y la
matriz superior en una variable llamada <code class="varname">U</code>.</p><p>Esto es la descomposición de
LU de una matriz también conocido como Crout y/o reducción de Cholesky. (ISBN 0-201-11577-8 pp.99-103) La
matriz triangular superior cuenta con una diagonal de valores 1 (uno). Esto no es el método de Doolittle en
las que los unos de la diagonal están sobre la matriz inferior.</p><p>No todas las matrices tienen la
descomposición de LU, por ejemplo <strong class="userinput"><code>[0,1;1,0]</code></strong> no lo hace y esta
función devuelve <code class="constant">false</code> en este caso, y establece <code class="varname">L</code>
y <code class="varname">U</code> a <code class="constant">null</code>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org
/LUDecomposition" target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/LUDecomposition.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Obtener el menor <code class="varname">i</code>-<code
class="varname">j</code> de una matriz.</p><p>Consulte <a class="ulink" href="http://planetmath.org/Minor";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Devolver las columnas que no son las columnas pivotes de una matriz.</p></dd><dt><span
class="term"><a name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm
(v,p...)</pre><p>Alias: <code class="function">norm</code></p><p>Obtener la norma p (o 2 normas si no se
suministra p) de
un vector.</p></dd><dt><span class="term"><a
name="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre class="synopsis">NullSpace
(T)</pre><p>Obtener el espacio nulo de una matriz. Ese es el núcleo de la aplicación lineal que representa la
matriz. Esto se devuelve como una matriz cuyo espacio de columna es el espacio nulo de <code
class="varname">T</code>.</p><p>Consulte <a class="ulink" href="http://planetmath.org/Nullspace";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre class="synopsis">Nullity (M)</pre><p>Alias: <code
class="function">nullity</code></p><p>Obtener la nulidad de una matriz. Es decir, devuelve la dimensión del
espacio nulo; la dimensión del núcleo de <code class="varname">M</code>.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/Nullity"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term">
<a name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre><p>Obtener el complemento ortogonal del espacio de
columnas.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Devuelve las columnas pivote de una matriz, que son columnas que tienen un 1 en la fila forma
reducida. También devuelve la fila en la que se producen.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Proyección del vector <code class="varname">v</code> sobre el sub-espacio <code
class="varname">W</code> con respecto al propio producto dado por <code class="varname">B</code>. Si <code
class="varname">B</code> no se da, entonces se usa el producto estándar Hermitiano. <code
class="varname">B</code> puede también ser una función sesquiline
al de dos argumentos o puede ser una matriz que devuelve una forma sesquilineal.</p></dd><dt><span
class="term"><a name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre
class="synopsis">QRDecomposition (A, Q)</pre><p>Obtener la descomposición QR de una matriz cuadrada <code
class="varname">A</code>, devuelve la matriz triangular superior <code class="varname">R</code> y establece
<code class="varname">Q</code> a la matriz ortogonal (unitaria). <code class="varname">Q</code> será una
referencia o <code class="constant">null</code> si no quiere que se devuelva ningún valor. Por ejemplo:
</p><pre class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>R =
QRDecomposition(A,&Q)</code></strong>
-</pre><p> tendrá la matriz triangular superior guardada en una variable llamada <code
class="varname">R</code> y la matriz ortogonal (unitaria) guardada en <code
class="varname">Q</code>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/QRDecomposition.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Devuelve el cociente de Rayleigh (también llamado el cociente
de Rayleigh-Ritz o ratio) de una matriz y un vector.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="
gel-function-RayleighQuotientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)</pre><p>Buscar valores propios de
<code class="varname">A</code> utilizando el método de iteración de cociente de Rayleigh. <code
class="varname">x</code> es una conjetura en un vector propio que será aleatoria. Esto tendrá una parte
imaginaria no nula si es posible encontrar valores propios complejos. El código ejecutará en la mayoría de
las interacciones <code class="varname">maxiter</code> y devuelve <code class="constant">null</code> si no se
puede obtener un error de <code class="varname">epsilon</code>. <code class="varname">vecref</code> será o
bién un <code class="constant">null</code> o una referencia a una variable donde se guarde el vector
propio.</p><p>Conuslte <a class="ulink" href="http://planetmath.org/RayleighQuotient";
target="_top">Planetmath</a> para obtener más información sobre el coci
ente de Rayleigh.</p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span></dt><dd><pre
class="synopsis">Rank (M)</pre><p>Alias: <code class="function">rank</code></p><p>Obtener el rango de una
matriz.</p><p>Consulte <a class="ulink" href="http://planetmath.org/SylvestersLaw";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Devolver la matriz de Rosser, que es un problemático y clásico test simétrico de valores
propios.</p></dd><dt><span class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre
class="synopsis">Rotation2D (ángulo)</pre><p>Alias: <code
class="function">RotationMatrix</code></p><p>Devolver la matriz correspondiente a la rotación alrededor del
origen en R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><
dd><pre class="synopsis">Rotation3DX (ángulo)</pre><p>Devuelve la matriz correspondiente a la rotación
alrededor del origen en R<sup>3</sup> sobre el eje x.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre class="synopsis">Rotation3DY
(ángulo)</pre><p>Devolver la matriz correspondiente a la rotación alrededor del origen en R<sup>3</sup> sobre
el eje Y.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(ángulo)</pre><p>Devolver la matriz correspondiente a la rotación alrededor del origen en R<sup>3</sup> sobre
el eje Z.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Obtener una matriz base para el espacio de filas de una
matriz.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre class="synopsi
s">SesquilinearForm (v,A,w)</pre><p>Evaluar (v,w) con respecto a la forma sesquilineal dada por la matriz
A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Devolver una función que evalúa dos vectores con
respecto a la forma sesquilineal dada por A.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Devuelve la forma normal de Smith de una matriz sobre los
campos (terminará con unos en la diagonal).</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Smith_normal_form"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>D
evuelve la forma normal de Smith para matrices cuadradas sobre enteros.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Smith_normal_form"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Resuelve el sistema lineal Mx=V, devuelve la
solución V si hay una única solución y <code class="constant">null</code> en cualquier otro caso.
Opcionalmente, se pueden usar dos parámetros de referencia para obtener M y V reducidos.</p></dd><dt><span
class="term"><a name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre
class="synopsis">ToeplitzMatrix (c, r...)</pre><p>Devuelve la matriz de Toeplitz que se construye con la
primera columna «c» y (opcionalmente) la primera fila «r». Si sólo se da la columna «c», entonces esta es
conjugada y la versión no conj
ugada la utiliza la primera fila para dar una matriz Hermitiana (si el primer elemento es
real).</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/ToeplitzMatrix";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Trace"></a>Trace</span></dt><dd><pre class="synopsis">Trace (M)</pre><p>Alias: <code
class="function">trace</code></p><p>Calcular la traza de una matriz. Esto es la suma de sus elementos
diagonales.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/Trace";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre class="synopsis">Transpose
(M)</pre><p>Traspuesta de una matriz. E
s lo mismo que el operador <strong class="userinput"><code>.'</code></strong>.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alias: <code class="function">vander</code></p><p>Devuelve la
matriz de Vandermonde.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Vandermonde_matrix"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>El ángulo de dos vectores con respecto al propio producto viene dado por <code
class="varname">B</code>. Si no se da <code class="varn
ame">B</code>, entonces se usará el producto estándar Hermitiano. <code class="varname">B</code> puede ser
una función sesquilineal de dos argumentos o bien, una matriz que devuelve una forma
sesquilineal.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Suma directa de los espacios vectoriales M y
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersección de subespacios dados por M y
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>La suma de los espacios vectoriales M y N, esto es {w |
w=m+n, m en M, n en N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd>
<pre class="synopsis">adj (m)</pre><p>Alias: <code class="function">Adjugate</code></p><p>Obtener el adjunto
clásico de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Alias: <code
class="function">CREF</code><code class="function">ColumnReducedEchelonForm</code></p><p>Calcular la forma en
escalón reducida por columnas.</p></dd><dt><span class="term"><a
name="gel-function-det"></a>det</span></dt><dd><pre class="synopsis">det (M)</pre><p>Alias: <code
class="function">Determinant</code></p><p>Obtener el determinante de una matriz.</p><p>Consulte la <a
class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Alias: <code c
lass="function">REF</code><code class="function">RowEchelonForm</code></p><p>Obtener la matriz escalonada
por fila. Es decir, aplicar la eliminación gausiana pero no hacer la reducción a <code
class="varname">M</code>. Las filas pivote están divididas para que todos los pivotes sean 1.</p><p>Consulte
la <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alias: <code class="function">RREF</code><code
class="function">ReducedRowEchelonForm</code></p><p>Obtener la matriz escalonada reducida por filas. Es
decir, aplicar la eliminación gausiana junto con la reducción a <code
class="varname">M</code>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Reduced_row_echelo
n_form" target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm";
target="_top">Planetmath</a> para obtener más información.</p></dd></dl></div></div><div
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class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-function-list-linear-algebra"></a>Álgebra
lineal</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Obtener la matriz auxiliar de tamaño <code
class="varname">n</code>. Esto es una matriz cuadrada que es toda ceros excepto la superdiagonal, que son
todos unos. Es la matriz de bloques de Jordan de un cero como valor propio.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> para obtener más información
sobre la forma canónica de Jordan.</p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm (v,A,w)
</pre><p>Evaluar (v,w) con respecto a la forma bilineal dada por la matriz A.</p></dd><dt><span
class="term"><a name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Devolver una función que evalúa dos vectores con respecto
a la forma bilineal dada por A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Alias: <code
class="function">CharPoly</code></p><p>Obtener el polinomio característico como un vector. Es decir, devuelve
los coeficientes del polinomio empezando por el término constante. Este polinomio se define por <strong
class="userinput"><code>det(M-xI)</code></strong>. Las raíces de este polinomio tienen como valor propio a
<code class="varname">M</code>. Consulte <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">Chara
cteristicPolynomialFunction</a>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Obtener el polinomio característico como una
función. Es decir, el polinomio se define por <strong class="userinput"><code>det(M-xI)</code></strong>. Las
raíces de este polinomio tienen un valor propio de <code class="varname">M</code>. Consulte <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> o <a
class="u
link" href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre class="synopsis">ColumnSpace
(M)</pre><p>Obtener una matriz base para el espacio de la columna de una matriz. Es decir, devuelve una
matriz la cual las columnas son las bases para el espacio de la columna <code class="varname">M</code>. Esto
es el espacio generado por las columnas de <code class="varname">M</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre
class="synopsis">CommutationMatrix (m, n)</pre><p>Devolver la matriz de conmutación <strong
class="userinput"><code>K(m,n)</code></strong> que es la única matriz <strong class="us
erinput"><code>m*n</code></strong> por <strong class="userinput"><code>m*n</code></strong> tal que <strong
class="userinput"><code>K(m,n) * MakeVector(A) = MakeVector(A.')</code></strong> para todas las matrices
<code class="varname">A</code> <code class="varname">m</code> por <code
class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre class="synopsis">CompanionMatrix
(p)</pre><p>Matriz acompañante de un polinomio (como vector).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Conjugada traspuesta de una matriz (adjunta). Es lo mismo que
el operador <strong class="userinput"><code>'</code></strong>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Conjugate_transpose"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/ConjugateTranspo
se" target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Alias: <code class="function">convol</code></p><p>Calcular la convolución de dos vectores
horizontales.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Calcular la convolución de dos vectores horizontales.
Devuelve el resultado como un vector y no se suman.</p></dd><dt><span class="term"><a
name="gel-function-CrossProduct"></a>CrossProduct</span></dt><dd><pre class="synopsis">CrossProduct
(v,w)</pre><p>Producto cruzado de dos vectores en R<sup>3</sup> como un vector columna.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Cross_product"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a n
ame="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre
class="synopsis">DeterminantalDivisorsInteger (M)</pre><p>Obtiene determinantes divisores de una matriz de
enteros.</p></dd><dt><span class="term"><a name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre
class="synopsis">DirectSum (M,N...)</pre><p>Suma directa de matrices.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-DirectSumMatrixVector"></a>DirectSumMatrixVector</span></dt><dd><pre
class="synopsis">DirectSumMatrixVector (v)</pre><p>Suma directa de un vector de matrices.</p><p>Consulte la
<a class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a>
para obtener más información.</p></dd><dt><span class="term"><a name="gel-function-Eigenvalues"></a>Eigenv
alues</span></dt><dd><pre class="synopsis">Eigenvalues (M)</pre><p>Alias: <code
class="function">eig</code></p><p>Obtener los valores propios de una matriz cuadrada. En la actualidad solo
funciona con matrices de tamaño 4 por 4 como máximo, o para matrices triangulares (cuyo valores propios están
en la diagonal).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Obtener los autovectores de una matriz cuadrada.
Opcionalmente, obtener los autovalores y su multiplicidad algebraica. Actualmente funciona sólo para matrices
de hasta 2x2.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Aplicar el proceso de Gram-Schmidt (a las columnas) con respecto al propio producto dado por
<code class="varname">B</code>. Si <code class="varname">B</code> no se da, entonces se utiliza el producto
Hermitiano estándar. <code class="varname">B</code> también puede ser una función sesquilineal de dos
argumentos o puede ser una matriz que devuelve una forma sesquilineal. Los vectores serán ortonormales con
respecto a <code class="varname">B</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/GramSchmidtOrthogonalization"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span></dt><dd><pre cl
ass="synopsis">HankelMatrix (c,r)</pre><p>La matriz de Hankel es una matriz cuyas diagonales (de izquierda a
derecha) son constantes. La primera fila es <code class="varname">c</code> y la última colúmna es <code
class="varname">r</code>. Se considera que ambos argumentos son vectores y que el último elemento de la fila
<code class="varname">c</code> es el mismo que el primer elemento de la columna <code
class="varname">r</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hankel_matrix"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Matriz de Hilbert de orden <code class="varname">n</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HilbertMatrix"; target="_top">Planet
math</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Image"></a>Image</span></dt><dd><pre class="synopsis">Image (T)</pre><p>Obtener la imagen
(espacio columna) de una transformación lineal.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre
class="synopsis">InfNorm (v)</pre><p>Obtener el operador norma de un vector, a veces también se denomina
norma suprema o norma máxima.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Obtiene los factores invariantes de una matriz cuadrada
de enteros.</p></dd><dt><span class="term"><a
name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatrix</span></dt><dd><pre clas
s="synopsis">InverseHilbertMatrix (n)</pre><p>Matriz inversa de Hilbert de orden <code
class="varname">n</code>.</p><p>Consulte la <a class="ulink"
href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Es una matriz Hermitian. Es decir, es igual a su traspuesta conjugada.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Hermitian_matrix"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://planetmath.org/HermitianMatrix"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsInSubspace"></a>IsInSubspace</span></dt><dd><pre class="synopsis">IsInSubspace
(v,W)</pre><p>Comprueba si un vector está en u
n subespacio.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Es una matriz (o número) invertible (La matriz de enteros es invertible si, y sólo si esta es
invertible sobre los enteros).</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Es una matriz (o un número) inversible sobre un
campo.</p></dd><dt><span class="term"><a name="gel-function-IsNormal"></a>IsNormal</span></dt><dd><pre
class="synopsis">IsNormal (M)</pre><p>Indica que <code class="varname">M</code> es una matriz normal. Es
decir, realiza <strong class="userinput"><code>M*M' == M'*M</code></strong>.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/NormalMatrix"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/NormalMatrix.html"; target="_top">Mathworld</a> par
a obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Indica que <code class="varname">M</code> es una matriz
definida positiva Hermitiana. Esto es si <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> es siempre estrictamente positivo para
cualquier vector <code class="varname">v</code>. <code class="varname">M</code> será cuadrada y Hermitiana
para ser definida positiva. La comprobación de que se lleva a cabo es que cada submatriz principal tiene un
determinante no negativo. (Consulte <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Tenga en cuenta que algunos
autores (por ejemplo Mathworld) no requieren que <code class="varname">M</code> sea Hermitiana, y entonces la
condición está en la parte real del propio producto, pero aquí no se compartirá este punto de vista
. Si quiere comprobarlo, hacer sólo la parte Hermitiana de la matriz <code class="varname">M</code> como
sigue: <strong class="userinput"><code>IsPositiveDefinite(M+M')</code></strong>.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Positive-definite_matrix"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Indica si <code class="varname">M</code> es una matriz
semidefinida positiva Hermitiana. Esto es si <strong
class="userinput"><code>HermitianProduct(M*v,v)</code></strong> es siempre no negativo para cualquier vector
<code class="varname">v</code>. <code class="va
rname">M</code> será cuadrada y Hermitiana para ser semidefinida positiva. La comprobación que se lleva a
cabo es que cada submatriz principal tenga un determinante no negativo. (Consulte <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Tenga en cuenta que algunos
autores no requieren que <code class="varname">M</code> sea Hermitiana, y entonces la condición está en la
parte real del propio producto, pero aquí no se compartirá este punto de vista. Si quiere comprobarlo, hacer
sólo la parte Hermitiana de la matriz <code class="varname">M</code> como sigue: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html"; target="_top">Mathworld</a> para obtener
más información.</p></dd><dt><span class="term"><a name=
"gel-function-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian
(M)</pre><p>Es matriz antihermítica. Esto es, la transposición conjugada es igual al negativo de la
matriz.</p><p>Consulte <a class="ulink" href="http://planetmath.org/SkewHermitianMatrix";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre class="synopsis">IsUnitary (M)</pre><p>¿Es
una matriz unitaria?. Esto es, hacer <strong class="userinput"><code>M'*M</code></strong> y <strong
class="userinput"><code>M*M'</code></strong> igual a la identidad.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/UnitaryTransformation"; target="_top">Planetmath</a> o <a class="ulink"
href="http://mathworld.wolfram.com/UnitaryMatrix.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-JordanBlock"></a>JordanBl
ock</span></dt><dd><pre class="synopsis">JordanBlock (n,lambda)</pre><p>Alias: <code
class="function">J</code></p><p>Obtener el bloque de Jordan correspondiente al valor propio <code
class="varname">lambda</code> con multiplicidad <code class="varname">n</code>.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> o <a
class="ulink" href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> para obtener
más información.</p></dd><dt><span class="term"><a name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre
class="synopsis">Kernel (T)</pre><p>Obtener el núcleo (espacio nulo) de una trasformación
lineal.</p><p>(Consulte <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Alias: <code class="function">T
ensorProduct</code></p><p>Calcula el producto de Kronecker (producto tensorial en base estándar) de dos
matrices.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>Obtener la descomposición de LU de <code class="varname">A</code> es decir, encontrar una
matriz triangular inferior y la matriz triangular superior cuyo producto es <code class="varname">A</code>.
Guarda el resultado en <code class="varname">L</code> y <code class="varname">U</code> que son referencias.
Devuelve <code class="constant">true</code> si se completó con éxito. Por ejemplo, suponga que «A» es una
matriz cuadrada, entonces después ejecute: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>LUDecomposition(A,&L,&U)</code></strong>
+</pre><p> tendrá la matriz inferior guardada en una variable llamada <code class="varname">L</code> y la
matriz superior en una variable llamada <code class="varname">U</code>.</p><p>Esto es la descomposición de
LU de una matriz también conocido como Crout y/o reducción de Cholesky. (ISBN 0-201-11577-8 pp.99-103) La
matriz triangular superior cuenta con una diagonal de valores 1 (uno). Esto no es el método de Doolittle en
las que los unos de la diagonal están sobre la matriz inferior.</p><p>No todas las matrices tienen la
descomposición de LU, por ejemplo <strong class="userinput"><code>[0,1;1,0]</code></strong> no lo hace y esta
función devuelve <code class="constant">false</code> en este caso, y establece <code class="varname">L</code>
y <code class="varname">U</code> a <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Obtener el menor <code class="varname">i</code>-<code
class="varname">j</code> de una matriz.</p><p>Consulte <a class="ulink" href="http://planetmath.org/Minor";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Devolver las columnas que no son las columnas pivotes de una matriz.</p></dd><dt><span
class="term"><a name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm
(v,p...)</pre><p>Alias: <code class="function">norm</code></p><p>Obtener la norma p (o 2 normas si no se
suministra p) de un vector.</p></dd><dt><span class="term"><a
name="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre class="synopsis">NullSpace
(T)</pre><p>Obtener el espacio nulo de una
matriz. Ese es el núcleo de la aplicación lineal que representa la matriz. Esto se devuelve como una matriz
cuyo espacio de columna es el espacio nulo de <code class="varname">T</code>.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/Nullspace"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre
class="synopsis">Nullity (M)</pre><p>Alias: <code class="function">nullity</code></p><p>Obtener la nulidad de
una matriz. Es decir, devuelve la dimensión del espacio nulo; la dimensión del núcleo de <code
class="varname">M</code>.</p><p>Consulte <a class="ulink" href="http://planetmath.org/Nullity";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre><p>Obtener el complemento ortogonal del esp
acio de columnas.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Devuelve las columnas pivote de una matriz, que son columnas que tienen un 1 en la fila forma
reducida. También devuelve la fila en la que se producen.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Proyección del vector <code class="varname">v</code> sobre el sub-espacio <code
class="varname">W</code> con respecto al propio producto dado por <code class="varname">B</code>. Si <code
class="varname">B</code> no se da, entonces se usa el producto estándar Hermitiano. <code
class="varname">B</code> puede también ser una función sesquilineal de dos argumentos o puede ser una matriz
que devuelve una forma sesquilineal.</p></dd><dt><span class="term"><a
name="gel-function-QRDecomposition"></a>QRDecomposition</span></
dt><dd><pre class="synopsis">QRDecomposition (A, Q)</pre><p>Obtener la descomposición QR de una matriz
cuadrada <code class="varname">A</code>, devuelve la matriz triangular superior <code
class="varname">R</code> y establece <code class="varname">Q</code> a la matriz ortogonal (unitaria). <code
class="varname">Q</code> será una referencia o <code class="constant">null</code> si no quiere que se
devuelva ningún valor. Por ejemplo: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>R = QRDecomposition(A,&Q)</code></strong>
+</pre><p> tendrá la matriz triangular superior guardada en una variable llamada <code
class="varname">R</code> y la matriz ortogonal (unitaria) guardada en <code class="varname">Q</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Devuelve el cociente de Rayleigh (también llamado el cociente
de Rayleigh-Ritz o ratio) de una matriz y un vector.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)</pre><p>Buscar valores propios de
<code class="varname">A</code> utilizando el método de iteración de cociente de Rayleigh. <code
class="varname">x</code> es una conjetura en un vector propio que será aleatoria. Esto tendrá una parte
imaginaria no nula si es posible encontrar valores propios complejos. El código ejecutará en la mayoría de
las interacciones <c
ode class="varname">maxiter</code> y devuelve <code class="constant">null</code> si no se puede obtener un
error de <code class="varname">epsilon</code>. <code class="varname">vecref</code> será o bién un <code
class="constant">null</code> o una referencia a una variable donde se guarde el vector propio.</p><p>Conuslte
<a class="ulink" href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> para obtener más
información sobre el cociente de Rayleigh.</p></dd><dt><span class="term"><a
name="gel-function-Rank"></a>Rank</span></dt><dd><pre class="synopsis">Rank (M)</pre><p>Alias: <code
class="function">rank</code></p><p>Obtener el rango de una matriz.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Devolver la matriz de Rosser,
que es un problemático y clásico test simétrico de valores propios.</p></dd><dt><span class="term"><a
name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(ángulo)</pre><p>Alias: <code class="function">RotationMatrix</code></p><p>Devolver la matriz correspondiente
a la rotación alrededor del origen en R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(ángulo)</pre><p>Devuelve la matriz correspondiente a la rotación alrededor del origen en R<sup>3</sup> sobre
el eje x.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre class="synopsis">Rotation3DY
(ángulo)</pre><p>Devolver la matriz correspondiente a la rotación alrededor del origen en R<sup>3</sup> sobre
el eje Y.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis"
Rotation3DZ (ángulo)</pre><p>Devolver la matriz correspondiente a la rotación alrededor del origen en
R<sup>3</sup> sobre el eje Z.</p></dd><dt><span class="term"><a
name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre class="synopsis">RowSpace (M)</pre><p>Obtener
una matriz base para el espacio de filas de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Evaluar (v,w) con respecto a la forma sesquilineal dada
por la matriz A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Devolver una función que evalúa dos vectores con
respecto a la forma sesquilineal dada por A.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre class="synopsis">Smit
hNormalFormField (A)</pre><p>Devuelve la forma normal de Smith de una matriz sobre los campos (terminará
con unos en la diagonal).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Devuelve la forma normal de Smith para matrices cuadradas
sobre enteros.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Resuelve el sistema lineal Mx=V, devuelve la
solución V si hay una única solución y <code class="constant">null</code> en cualquier otro caso.
Opcionalmente, se pueden usar dos parámetros de referencia para obtener M y V reducidos.</p></dd><dt><span
class="term"><a name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre
class="synopsis">ToeplitzMatrix (c, r...)</pre><p>Devuelve la matriz de Toeplitz que se construye con la
primera columna «c» y (opcionalmente) la primera fila «r». Si sólo se da la columna «c», entonces esta es
conjugada y la versión no conjugada la utiliza la primera fila para dar una matriz Hermitiana (si el primer
elemento es real).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Alias: <code class="function">trace</code></p><p>Calcular la traza de una
matriz. Esto es la suma de sus elementos diagonales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Traspuesta de una matriz. Es lo mismo que el operador <strong
class="userinput"><code>.'</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alias: <code class="function">vander</code></p><p>Devuelve la
matriz de Vandermonde.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>El ángulo de dos vectores con respecto al propio producto viene dado por <code
class="varname">B</code>. Si no se da <code class="varname">B</code>, entonces se usará el producto estándar
Hermitiano. <code class="varname">B</code> puede ser una función sesquilineal de dos argumentos o bien, una
matriz que devuelve una forma sesquilineal.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Suma directa de los espacios vectoriales M y
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersección de subespacios dados por M y
N.</p></dd><dt><span class="ter
m"><a name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>La suma de los espacios vectoriales M y N, esto es {w |
w=m+n, m en M, n en N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Alias: <code class="function">Adjugate</code></p><p>Obtener el adjunto
clásico de una matriz.</p></dd><dt><span class="term"><a
name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Alias: <code
class="function">CREF</code><code class="function">ColumnReducedEchelonForm</code></p><p>Calcular la forma en
escalón reducida por columnas.</p></dd><dt><span class="term"><a
name="gel-function-det"></a>det</span></dt><dd><pre class="synopsis">det (M)</pre><p>Alias: <code
class="function">Determinant</code></p><p>Obtener el determinante de una matriz.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
+ </p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Alias: <code class="function">REF</code><code
class="function">RowEchelonForm</code></p><p>Obtener la matriz escalonada por fila. Es decir, aplicar la
eliminación gausiana pero no hacer la reducción a <code class="varname">M</code>. Las filas pivote están
divididas para que todos los pivotes sean 1.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alias: <code class="function">RREF</code><code
class="function">ReducedRowEchelonForm</code></p><p>Obtener la matriz escalonada reducida por filas. Es
decir, aplicar la eliminación gausiana junto con la reducción a <code class="varname">M</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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-<html><head><meta http-equiv="Content-Type" content="text/html;
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V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
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href="ch11s11.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style=
"clear: both"><a name="genius-gel-function-list-combinatorics"></a>Combinatoria</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Obtener el
<code class="varname">n</code>-ésimo número de Catalan.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Obtener todas las combinaciones de «k» números desde 1 a «n» como un vector de vectores.
(Consulte <a class="link"
href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Doble factoria
l: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>Consulte <a class="ulink"
href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre class="synopsis">Factorial
(n)</pre><p>Factorial: <strong class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>Consulte <a
class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre
class="synopsis">FallingFactorial (n,k)</pre><p>Factorial descendente: <strong class="userinput"><code>(n)_k
= n(n-1)...(n-(k-1))</code></strong></p><p>Consulte la <a class="ulink"
href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="ge
l-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre class="synopsis">Fibonacci (x)</pre><p>Alias: <code
class="function">fib</code></p><p>Calcular el <code class="varname">n</code>-ésimo número de Fibonacci. El
número se define recursivamente por <strong class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) +
Fibonacci(n-2)</code></strong> y <strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) =
1</code></strong>.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/FibonacciSequence";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>Calcular el número de Frobenius. Calcular en n�
�mero más pequeño que no se puede dar como una combinación de entero lineal no negativo de un vector dado de
enteros no negativos. El vector se puede dar como números separados o un simple vector. Todos los números
tendrán un máximo común divisor de enteros «GCD» de 1.</p><p>Consulte la <a class="ulink"
href="http://mathworld.wolfram.com/FrobeniusNumber.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(regla_de_combinación)</pre><p>Matriz de Galois dada una regla de combinación lineal
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Buscar el vector <code class="varname">c</code> de enteros no negativos de tal manera que al
realizar el producto escalar con <code class="varname"
v</code> es igual a n. Si no es posible, se devuelve <code class="constant">null</code>. <code
class="varname">v</code> estará ordenada de forma incremental y estará constituida de enteros no
negativos.</p><p>Consulte la <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alias: <code class="function">HarmonicH</code></p><p>Número harmónico, el <code
class="varname">n</code>-ésimo número harmónico de orden <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre
class="synopsis">Hofstadter (n)</pre><p>Función q(n) de Hofstadter definida por q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a name="gel-function-LinearRecursiveSequenc
e"></a>LinearRecursiveSequence</span></dt><dd><pre class="synopsis">LinearRecursiveSequence
(seed_values,combining_rule,n)</pre><p>Calcular la sucesión lineal recursiva utilizando el escalamiento de
Galois.</p></dd><dt><span class="term"><a name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre
class="synopsis">Multinomial (v,arg...)</pre><p>Calcular los coeficientes multinomiales. Toma un vector de
<code class="varname">k</code> enteros no negativos y calcula el coeficiente multinomial. Esto corresponde al
coeficiente en el polinomio homogéneo en <code class="varname">k</code> variables con las correspondientes
potencias.</p><p>La fórmula para <strong class="userinput"><code>Multinomial(a,b,c)</code></strong> se puede
escribir como: </p><pre class="programlisting">(a+b+c)! / (a!b!c!)
-</pre><p>. En otras palabras, si sólo hay dos elementos, entonces <strong
class="userinput"><code>Multinomial(a,b)</code></strong> es lo mismo que <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> o <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Multinomial_theorem"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, o <a class="ulink"
href="http://mathworld.wolfram.com/MultinomialCoefficient.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Obtener las combinaciones que v devolverá después de su ejecución. La primera combinación será
<strong class="userinput"><code>[1:k]</code></strong>. Esta función es útil si tiene muchas c
ombinaciones que pasar y no quiere olvidarse de guardarlas todas.</p><p>Por ejemplo, con «Combinations»
normalmente escribiría un bucle como sigue: </p><pre class="screen"><strong class="userinput"><code>for n in
Combinations (4,6) do (
+<html><head><meta http-equiv="Content-Type" content="text/html;
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header"><tr><th colspan="3" align="center">Combinatoria</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s09.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s11.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style=
"clear: both"><a name="genius-gel-function-list-combinatorics"></a>Combinatoria</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Obtener el
<code class="varname">n</code>-ésimo número de Catalan.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Obtener todas las combinaciones de «k» números desde 1 a «n» como un vector de vectores.
(Consulte <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Doble factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>Consulte <a
class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre class="synopsis">Factorial
(n)</pre><p>Factorial: <strong class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>Consulte <a
class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre
class="synopsis">FallingFactorial (n,k)</pre><p>Factorial descendente: <strong class="userinput"><code>(n)_k
= n(n-1)...(n-(k-1))</code></strong></p><p>Consu
lte la <a class="ulink" href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre class="synopsis">Fibonacci (x)</pre><p>Alias:
<code class="function">fib</code></p><p>Calcular el <code class="varname">n</code>-ésimo número de Fibonacci.
El número se define recursivamente por <strong class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) +
Fibonacci(n-2)</code></strong> y <strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) =
1</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
+ Calculate the Frobenius number. That is calculate largest
+ number that cannot be given as a non-negative integer linear
+ combination of a given vector of non-negative integers.
+ The vector can be given as separate numbers or a single vector.
+ All the numbers given should have GCD of 1.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(regla_de_combinación)</pre><p>Matriz de Galois dada una regla de combinación lineal
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Buscar el vector <code class="varname">c</code> de enteros no negativos de tal manera que al
realizar el producto escalar con <code class="varname">v</code> es igual a n. Si no es posible, se devuelve
<code class="constant">null</code>. <code class="varname">v</code> estará ordenada de forma incremental y
estará constituida de enteros no negativos.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alias: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Función q(n) de Hofstadter definida por q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Calcular la sucesión lineal
recursiva utilizando el escalamiento de Galois.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calcular los coeficientes multinomiales. Toma un vector de <code class="varname">k</code>
enteros no negativos y calcula el coeficiente multinomial. Esto corresponde al coeficiente en el polinomio
homogéneo en <code class="varname">k</code> variables con las correspondientes potencias.</p><p>La fórmula
para <strong class="userinput"><code>Multinomial(a,b,c)</code></strong> se puede escribir como: </p><pre
class="programlisting">(a+b+c)! / (a!b!c!)
+</pre><p>. En otras palabras, si sólo hay dos elementos, entonces <strong
class="userinput"><code>Multinomial(a,b)</code></strong> es lo mismo que <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> o <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Obtener las combinaciones que v devolverá después de su ejecución. La primera combinación será
<strong class="userinput"><code>[1:k]</code></strong>. Esta función es útil si tiene muchas combinaciones que
pasar y no quiere olvidarse de guardarlas todas.</p><p>Por ejemplo, con «Combinations» normalmente escribiría
un bucle como sigue: </p><pre class="screen"><strong class="userinput"><code>for n in Combinations (4,6) do (
AlgunaFuncion (n)
);</code></strong>
</pre><p> Pero con «NextCombination» escribiría algo como lo siguiente: </p><pre class="screen"><strong
class="userinput"><code>n:=[1:4];
do (
AlgunaFuncion (n)
) while not IsNull(n:=NextCombination(n,6));</code></strong>
-</pre><p> Consulte también <a class="link"
href="ch11s10.html#gel-function-Combinations">Combinations</a>.</p></dd><dt><span class="term"><a
name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre class="synopsis">Pascal (i)</pre><p>Obtener el
triángulo de Pascal como una matriz. Esto devolverá una <code class="varname">i</code>+1 por <code
class="varname">i</code>+1 la diagonal inferior de la matriz que es el triángulo de Pascal después de <code
class="varname">i</code> iteraciones.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Obtener todas las permutaciones de <code class="varname">k</code> números desde el 1 al <code
class="varname">n</code> como un vector de vectores.</p><p>Consulte <a class="ulink"
href="http://mathworld.wolf
ram.com/Permutation.html" target="_top">Mathworld</a> o la <a class="ulink"
href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alias: <code class="function">Pochhammer</code></p><p>(Puchhammer) factorial creciente: (n)_k =
n(n+1)...(n+(k-1)).</p><p>Consulte <a class="ulink" href="http://planetmath.org/RisingFactorial";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alias: <code
class="function">StirlingS1</code></p><p>Número de Stirling de primera clase.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/StirlingNumbersOfTheFirstKind"; target="_top">Planetmath</a> o <a
class="
ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html"; target="_top">Mathworld</a>
para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Alias: <code
class="function">StirlingS2</code></p><p>Número de Stirling de segunda clase.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/StirlingNumbersSecondKind"; target="_top">Planetmath</a> o <a
class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangular (
nth)</pre><p>Calcular el <code class="varname">n</code>-ésimo número triangular.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/TriangularNumbers"; target="_top">Planetmath</a>> para obtener
más información.</p></dd><dt><span class="term"><a name="gel-function-nCr"></a>nCr</span></dt><dd><pre
class="synopsis">nCr (n,r)</pre><p>Alias: <code class="function">Binomial</code></p><p>Calcular
combinaciones, es decir, el coeficiente del binomio. <code class="varname">n</code> puede ser cualquier
número real.</p><p>Consulte <a class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a>
para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-nPr"></a>nPr</span></dt><dd><pre class="synopsis">nPr (n,r)</pre><p>Calcular el número de
permutaciones de tamaño <code class="varname">r</code> de números desde el 1 al <code
class="varname">n</code>.</p><p>Consulte <a class="ulink" href="http://mathworld.wolfram.com/Permutat
ion.html" target="_top">Mathworld</a> o la <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation";
target="_top">Wikipedia</a> para obtener más información.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s09.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s11.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Álgebra lineal
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%"
align="right" valign="top"> Cálculo</td></tr></table></div></body></html>
+</pre><p> Consulte también <a class="link"
href="ch11s10.html#gel-function-Combinations">Combinations</a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Obtener el triángulo de Pascal como una matriz. Esto devolverá una <code
class="varname">i</code>+1 por <code class="varname">i</code>+1 la diagonal inferior de la matriz que es el
triángulo de Pascal después de <code class="varname">i</code> iteraciones.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Obtener todas las permutaciones de <code class="varname">k</code> números desde el 1 al <code
class="varname">n</code> como un vector de vectores.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alias: <code class="function">Pochhammer</code></p><p>(Puchhammer) factorial creciente: (n)_k =
n(n+1)...(n+(k-1)).</p><p>Consulte <a class="ulink" href="http://planetmath.org/RisingFactorial";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alias: <code
class="function">StirlingS1</code></p><p>Número de Stirling de primera clase.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/StirlingNumbersOfTheFirstKind"; target="_top">Planetmath</a> o <a
class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a name="gel
-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Alias: <code
class="function">StirlingS2</code></p><p>Número de Stirling de segunda clase.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/StirlingNumbersSecondKind"; target="_top">Planetmath</a> o <a
class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangular
(nth)</pre><p>Calcular el <code class="varname">n</code>-ésimo número triangular.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/TriangularNumbers"; target="_top">Plan
etmath</a>> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-nCr"></a>nCr</span></dt><dd><pre class="synopsis">nCr (n,r)</pre><p>Alias: <code
class="function">Binomial</code></p><p>Calcular combinaciones, es decir, el coeficiente del binomio. <code
class="varname">n</code> puede ser cualquier número real.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/Choose"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre
class="synopsis">nPr (n,r)</pre><p>Calcular el número de permutaciones de tamaño <code
class="varname">r</code> de números desde el 1 al <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">Siguiente</a></td></tr><tr><td width="40%" align="left"
valign="top">Álgebra lineal </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top">
Cálculo</td></tr></table></div></body></html>
diff --git a/help/es/html/ch11s11.html b/help/es/html/ch11s11.html
index c794ec8..ba1ac2e 100644
--- a/help/es/html/ch11s11.html
+++ b/help/es/html/ch11s11.html
@@ -1 +1,29 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Cálculo</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manual
de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de funciones GEL"><link rel="prev"
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bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
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href="ch11s12.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: bo
th"><a name="genius-gel-function-list-calculus"></a>Cálculo</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integrar f usando la Regla Compuesta de Simpson en
el intervalo [a,b] con n subintervalos y un error de max(f'''')*h^4*(b-a)/180, n debe ser
entero.</p><p>Consulte <a class="ulink" href="http://planetmath.org/SimpsonsRule";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)</pre><p>Integración
de F por la Regla compuesta de Simpson en el intervalo [a,b] con el número de pasos calculado por la cuarta
derivada y la tolerancia deseada.</p><p
Consulte <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Intentar calcular la derivada, primero simbólicamente y después
numéricamente.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Devolver una función que es una extensión periódica
par de <code class="function">f</code> con medio periodo <code class="varname">L</code>. Esto es una
función que se define en el intervalo <strong class="userinput"><code>[0,L]</code></strong> extendido para
ser par en <strong class="userinput"><code>[-L,L]
</code></strong> y entonces extendido para ser periódico con periodo <strong
class="userinput"><code>2*L</code></strong>.</p><p>Consulte <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> y <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Desde la versión 1.0.7 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Devuelve una función que es una serie de Fourier con
coeficientes devueltos por los vectores <code class="varname">a</code> (senos) y <code
class="varname">b</code> (cosenos). Tenga en cuenta que <strong class="userinput"><code>a@(1)</code></strong>
es el coeficiente constante. Es decir, <strong class="userinput"><code>a@(n)</code></strong> se refiere al
término <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, mientras q
ue <strong class="userinput"><code>b@(n)</code></strong> se refiere al término <strong
class="userinput"><code>sin(x*n*pi/L)</code></strong>. Tanto <code class="varname">a</code> o <code
class="varname">b</code> puede ser <code class="constant">null</code>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,inicio,inc)</pre><p>Intenta calcular un producto infinito para una función de un sólo
parámetro.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,inicio,inc)</pre><p>Intenta calcular un producto infinito para
una función
de dos parámetros con func(arg,n)</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,inicio,inc)</pre><p>Intentar calcular una suma infinita para una función de un sólo
parámetro.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre class="synopsis">InfiniteSum2
(func,arg,inicio,inc)</pre><p>Intenta calcular una suma infinita para una función de dos parámetros con
func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Comprueba si una función real es continua en x0 calculando el límite en ese
punto.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Comprobar la diferenciabilidad aproximando los límites
izquierdo
y derecho y comparándolos.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calcular el límite por la izquierda de una función real en x0.</p></dd><dt><span
class="term"><a name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit
(f,x0)</pre><p>Calcular el límite de una función real en x0. Intenta calcular tanto el límite por la
derecha como por la izquierda.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Integración por la regla del punto medio.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alias: <code
class="function">NDerivative</code></p><p>Intentar calcular la derivada numérica.</p><p>Consulte la <a
class="ulink" href
="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Devuelve un vector de vectores <strong
class="userinput"><code>[a,b]</code></strong> donde <code class="varname">a</code> son los coeficientes
cosenos y <code class="varname">b</code> son los coeficientes senos de la serie de Fourier de <code
class="function">f</code> con medio periodo <code class="varname">L</code> (esto se define en <strong
class="userinput"><code>[-L,L]</code></strong> y extendido periódicamente) con coeficientes hasta <code
class="varname">N</code>-ésimo harmónico calculado numéricamente. Los coeficientes se calculan por la
integración numérica al usar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="fun
ction">NumericalIntegral</code></a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> para obtener más
información.</p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Devuelve una función que es la serie de
Fourier de <code class="function">f</code> con medio periodo <code class="varname">L</code> (esto se define
en <strong class="userinput"><code>[-L,L]</code></strong> y extendido periódicamente) con coeficientes hasta
<code class="varname">N</code>-ésimo harmónico calculado numéricamente. Esto es, la serie trigonométrica real
compuesta de senos y cosenos. Los coeficientes se calculan por la integración numé
rica al utilizar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> para obtener más
información.</p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Devuelve un vector de coeficientes
de coseno de la serie de Fourier de <code class="function">f</code> con medio periodo <code
class="varname">L</code>. Es decir, se toma <code class="function">f</code> definida en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica par y calcula la serie de Fo
urier, que sólo tiene cosenos como términos. La serie se calcula hasta la <code
class="varname">N</code>-ésima harmónica. Los coeficientes se calculan por la integración numérica al
utilizar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>. Tenga en cuenta que <strong
class="userinput"><code>a@(1)</code></strong> es el coeficiente constante. Es decir, <strong
class="userinput"><code>a@(n)</code></strong> se refiere a el término <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> para obtener más
información.</p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosin
eSeriesFunction</span></dt><dd><pre class="synopsis">NumericalFourierCosineSeriesFunction
(f,L,N)</pre><p>Devuelve una función que es el coseno de la serie de Fourier de <code
class="function">f</code> con medio periodo <code class="varname">L</code>. Es decir, se toma <code
class="function">f</code> definida en <strong class="userinput"><code>[0,L]</code></strong> toma la extensión
periódica par y calcula la serie de Fourier, que sólo tiene coseno como términos. La serie se calcula hasta
la <code class="varname">N</code>-ésima harmónica. Los coeficientes se calculan por la integración numérica
al utilizar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> para obtener más
inform
ación.</p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Devuelve un vector de coeficientes
de senos de la serie de Fourier de <code class="function">f</code> con medio periodo <code
class="varname">L</code>. Es decir, se toma <code class="function">f</code> definido en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica impar y calcula la serie de
Fourier, que sólo tiene senos como términos. La serie se calcula hasta el <code
class="varname">N</code>-ésimo harmónico. Los coeficientes se calculan por la integración numérica al
utilizar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wi
ki/Fourier_series" target="_top">Wikipedia</a> o <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> para obtener más
información.</p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Devuelve una función que es el seno de
la serie de Fourier de <code class="function">f</code> con medio periodo <code class="varname">L</code>. Es
decir, se toma <code class="function">f</code> definida en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica impar y calcula ls series de
Fourier, que sólo tiene seno como términos. La serie se calcula hasta la <code class="varname">N</code>-ésima
harmónica. Los coeficientes se calculan por la integración numérica al utilizar <a class="link" href="ch11s11
.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>Consulte la
<a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> o <a
class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> para
obtener más información.</p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integración por el conjunto de reglas en
NumericalIntegralFunction de f desde «a» a «b» usando NumericalIntegralSteps pasos.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Intentar calcular la derivada numérica por la
izquierda.</p></dd><dt><span class="term"><a name="gel-function-NumericalLimi
tAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre class="synopsis">NumericalLimitAtInfinity
(_f,step_fun,tolerance,successive_for_success,N)</pre><p>Intentar calcular el límite de f(step_fun(i)), para
i desde 1 hasta N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Intentar calcular la derivada numérica por la
derecha.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Devuelve una función que es la extensión periódica impar
de <code class="function">f</code> con medio periodo <code class="varname">L</code>. Esto es una función
definida en el intervalo <strong class="userinput"><code>[0,L]</code></strong> extendida para ser impar en
<strong class="userinput"><code>[-L,L]</code></strong> y entonces ex
tendida para ser periódica con periodo <strong class="userinput"><code>2*L</code></strong>.</p><p>Consulte
también <a class="link" href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> y <a
class="link" href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Desde la versión
1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Calcular la derivada de un lado usando una
fórmula de 5 puntos.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Calcular la derivada de un lado usando una
fórmula de tres puntos.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre class="synopsi
s">PeriodicExtension (f,a,b)</pre><p>Devuelve una función que es la extensión periódica de <code
class="function">f</code> que se define en el intervalo <strong class="userinput"><code>[a,b]</code></strong>
y tiene un periodo <strong class="userinput"><code>b-a</code></strong>.</p><p>Consulte también <a
class="link" href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> y <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>Desde la versión
1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre class="synopsis">RightLimit
(f,x0)</pre><p>Calcular el límite por la derecha de una función real en x0.</p></dd><dt><span
class="term"><a name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Calcular la derivada de dos lados usando una fór
mula de cinco puntos.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Calcular la derivada de dos lados usando una
fórmula de tres puntos.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Cálculo</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manual
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width="20%" align="left"><a accesskey="p" href="ch11s10.html">Anterior</a> </td><th width="60%"
align="center">Capítulo 11. Lista de funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
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class="titlepage"><div><div><h2 class="title" style="clear: bo
th"><a name="genius-gel-function-list-calculus"></a>Cálculo</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integrar f usando la Regla Compuesta de Simpson en
el intervalo [a,b] con n subintervalos y un error de max(f'''')*h^4*(b-a)/180, n debe ser
entero.</p><p>Consulte <a class="ulink" href="http://planetmath.org/SimpsonsRule";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)</pre><p>Integración
de F por la Regla compuesta de Simpson en el intervalo [a,b] con el número de pasos calculado por la cuarta
derivada y la tolerancia deseada.</p><p
Consulte <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Intentar calcular la derivada, primero simbólicamente y después
numéricamente.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Derivative";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Devolver una función que es una extensión periódica
par de <code class="function">f</code> con medio periodo <code class="varname">L</code>. Esto es una
función que se define en el intervalo <strong class="userinput"><code>[0,L]</code></strong> extendido para
ser par en <strong class="userinput"><code>[-L,L]
</code></strong> y entonces extendido para ser periódico con periodo <strong
class="userinput"><code>2*L</code></strong>.</p><p>Consulte <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> y <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Desde la versión 1.0.7 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Devuelve una función que es una serie de Fourier con
coeficientes devueltos por los vectores <code class="varname">a</code> (senos) y <code
class="varname">b</code> (cosenos). Tenga en cuenta que <strong class="userinput"><code>a@(1)</code></strong>
es el coeficiente constante. Es decir, <strong class="userinput"><code>a@(n)</code></strong> se refiere al
término <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, mientras q
ue <strong class="userinput"><code>b@(n)</code></strong> se refiere al término <strong
class="userinput"><code>sin(x*n*pi/L)</code></strong>. Tanto <code class="varname">a</code> o <code
class="varname">b</code> puede ser <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,inicio,inc)</pre><p>Intenta calcular un producto infinito para una función de un sólo
parámetro.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,inicio,inc)</pre><p>Intenta calcular un producto infinito para
una función de dos parámetros con func(arg,n)</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,inicio,inc)</pre><p>Intentar calcular una suma infinita para una función de un sólo
parámetro.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre class="synopsis">InfiniteSum2
(func,arg,inicio,inc)</pre><p>Intenta calcular una suma infinita para una función de do
s parámetros con func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Comprueba si una función real es continua en x0 calculando el límite en ese
punto.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Comprobar la diferenciabilidad aproximando los límites
izquierdo y derecho y comparándolos.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calcular el límite por la izquierda de una función real en x0.</p></dd><dt><span
class="term"><a name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit
(f,x0)</pre><p>Calcular el límite de una función real en x0. Intenta calcular tanto el límite por la
derecha como por la izquierda.</p></dd><
dt><span class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre
class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integración por la regla del punto medio.</p></dd><dt><span
class="term"><a name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alias: <code
class="function">NDerivative</code></p><p>Intentar calcular la derivada numérica.</p><p>Consulte la <a
class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Devuelve un vector de vectores <strong
class="userinput"><code>[a,b]</code></strong> donde <code class="varname">a</code> son los coeficientes
cosenos y <code class="varname">b
</code> son los coeficientes senos de la serie de Fourier de <code class="function">f</code> con medio
periodo <code class="varname">L</code> (esto se define en <strong
class="userinput"><code>[-L,L]</code></strong> y extendido periódicamente) con coeficientes hasta <code
class="varname">N</code>-ésimo harmónico calculado numéricamente. Los coeficientes se calculan por la
integración numérica al usar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Devuelve una función que es la serie de
Fourier de <code class="function">f</code> con medio periodo <code class="varname">L</code> (esto se define
en <strong class="userinput"><code>[-L,L]</code></strong> y extendido periódicamente) con coeficientes hasta
<code class="varname">N</code>-ésimo harmónico calculado numéricamente. Esto es, la serie trigonométrica real
compuesta de senos y cosenos. Los coeficientes se calculan por la integración numérica al utilizar <a
class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Devuelve un vector de coeficientes
de coseno de la serie de Fourier de <code class="function">f</code> con medio periodo <code
class="varname">L</code>. Es decir, se toma <code class="function">f</code> definida en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica par y calcula la serie de Fourier,
que sólo tiene cosenos como términos. La serie se calcula hasta la <code class="varname">N</code>-ésima
harmónica. Los coeficientes se calculan por la integración numérica al utilizar <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>. Tenga
en cuenta que <strong class="userinput"><code>a@(1)
</code></strong> es el coeficiente constante. Es decir, <strong
class="userinput"><code>a@(n)</code></strong> se refiere a el término <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Devuelve una función que es el coseno
de la serie de Fourier de <code class="function">f</code> con medio periodo <code class="varname">L</code>.
Es decir, se toma <code class="function">f</code> definida en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica par y calcula la serie de Fourier,
que sólo tiene coseno como términos. La serie se calcula hasta la <code class="varname">N</code>-ésima
harmónica. Los coeficientes se calculan por la integración numérica al utilizar <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Devuelve un vector de coeficientes
de senos de la serie de Fourier de <code class="function">f</code> con medio periodo <code
class="varname">L</code>. Es decir, se toma <code class="function">f</code> definido en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica impar y calcula la serie de
Fourier, que sólo tiene senos como términos. La serie se calcula hasta el <code
class="varname">N</code>-ésimo harmónico. Los coeficientes se calculan por la integración numérica al
utilizar <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Devuelve una función que es el seno de
la serie de Fourier de <code class="function">f</code> con medio periodo <code class="varname">L</code>. Es
decir, se toma <code class="function">f</code> definida en <strong
class="userinput"><code>[0,L]</code></strong> toma la extensión periódica impar y calcula ls series de
Fourier, que sólo tiene seno como términos. La serie se calcula hasta la <code class="varname">N</code>-ésima
harmónica. Los coeficientes se calculan por la integración numérica al utilizar <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Desde la versión 1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integración por el conjunto de reglas en
NumericalIntegralFunction de f desde «a» a «b» usando NumericalIntegralSteps pasos.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Intentar calcular la derivada numérica por la
izquierda.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Intentar
calcular el límite de f(step_fun(i)), para i desde 1 hasta N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightD
erivative</span></dt><dd><pre class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Intentar calcular la
derivada numérica por la derecha.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Devuelve una función que es la extensión periódica impar
de <code class="function">f</code> con medio periodo <code class="varname">L</code>. Esto es una función
definida en el intervalo <strong class="userinput"><code>[0,L]</code></strong> extendida para ser impar en
<strong class="userinput"><code>[-L,L]</code></strong> y entonces extendida para ser periódica con periodo
<strong class="userinput"><code>2*L</code></strong>.</p><p>Consulte también <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> y <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Desde la versión 1.
0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</span></dt><dd><pre
class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Calcular la derivada de un lado usando una
fórmula de 5 puntos.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Calcular la derivada de un lado usando una
fórmula de tres puntos.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Devuelve una función que es la extensión periódica de
<code class="function">f</code> que se define en el intervalo <strong
class="userinput"><code>[a,b]</code></strong> y tiene un periodo <strong
class="userinput"><code>b-a</code></strong>.</p><p>Consulte también <a class="link" href="ch11
s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> y <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>Desde la versión
1.0.7 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre class="synopsis">RightLimit
(f,x0)</pre><p>Calcular el límite por la derecha de una función real en x0.</p></dd><dt><span
class="term"><a name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Calcular la derivada de dos lados usando una
fórmula de cinco puntos.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Calcular la derivada de dos lados usando una
fórmula de tres puntos.</p></dd></dl></div></div><div class="navfooter"><
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class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-gel-function-list-functions"></a>Funciones</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Alias:
<code class="function">Arg</code><code class="function">arg</code></p><p>argumento (ángulo) de un número
complejo.</p></dd><dt><span class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre
class="synopsis">BesselJ0 (x)</pre><p>Función de Bessel de primer tipo de orden 0. Implementada solo para
números reales.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.16 en
adelante.</p></dd><dt><span class="term"><a name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre
class="synopsis">BesselJ1 (x)</pre><p>Función de Bessel de primer tipo de orde
n 1. Implementada solo para números reales.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> para obtener más
información.</p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Función
de Bessel de primer tipo de orden <code class="varname">n</code>. Implementada solo para números
reales.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.16 en
adelante.</p></dd><dt><span class="term"><a name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre
class="synopsis">BesselY0 (x)</pre><p>Función de Bessel de segundo tipo de orden 0. Implementada solo para
números reales.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.16 en
adelante.</p></dd><dt><span class="term"><a name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre
class="synopsis">BesselY1 (x)</pre><p>Función de Bessel de segunto tipo de orden 1. Implementada solo para
números reales.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.16 en
adelante.</p></dd><dt><span class="term"><a name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre
class="synopsis">BesselYn (n,x)</pre><p>Función de Bessel de segundo tipo de orden <code
class="varname">n</code>. Implementada solo para números reales.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> para obtener más
información.</p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="ter
m"><a name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre
class="synopsis">DirichletKernel (n,t)</pre><p>Núcleo de Dirichlet de orden <code
class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Devuelve 1 si y sólo si todos los elementos son cero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alias: <code class="function">erf</code></p><p>La función de error, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Error_function";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/ErrorFunction";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-FejerKernel"></a>FejerKernel</span></dt><dd><pre c
lass="synopsis">FejerKernel (n,t)</pre><p>Núcleo de Fejer de orden <code class="varname">n</code> evaluado
en <code class="varname">t</code></p><p>Consulte <a class="ulink" href="http://planetmath.org/FejerKernel";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alias: <code class="function">Gamma</code></p><p>La función «Gamma». Actualmente sólo
implementada para valores reales.</p><p>Consulte <a class="ulink" href="http://planetmath.org/GammaFunction";
target="_top">Planetmath</a> o <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Devuelve 1 si y sólo si todos los elementos son igual
es.</p></dd><dt><span class="term"><a name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre
class="synopsis">LambertW (x)</pre><p>La rama principal de la función de Lambert W calculada sólo para los
valores reales más grandes o iguales que <strong class="userinput"><code>-1/e</code></strong>. Es decir, que
la función <code class="function">LambertW</code> es la inversa de la expresión <strong
class="userinput"><code>x*e^x</code></strong>. Incluso para una variable real <code class="varname">x</code>
esta expresión no es uno a uno y por lo tanto tiene dos ramas más <strong
class="userinput"><code>[-1/e,0)</code></strong>. Consulte <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> para otras ramas
reales.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> para más información.</p><p>Desde la versión 1.0.18 en
adelante.</p></dd><dt><span cl
ass="term"><a name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1
(x)</pre><p>La rama menos uno «-1» de la función de Lambert W calculada sólo para valores reales más grandes
o igual a <strong class="userinput"><code>-1/e</code></strong> y menor que 0. Es decir, <code
class="function">LambertWm1</code> es la segunda rama de la inversa de <strong
class="userinput"><code>x*e^x</code></strong>. Consulte <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> para la rama
principal.</p><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> para más información.</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Buscar el primer valor donde
f(x)=0.</p></dd><dt><span class="term"><a name="gel-function-MoebiusDiskMa
pping"></a>MoebiusDiskMapping</span></dt><dd><pre class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Mapa de
Moebius del disco a sí mismo mapeando a 0.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Mapa de Moebius usando el radio cruzado z2,z3,z4 a 1,0 e infinito
respectivamente.</p><p>Consulte <a class="ulink" href="http://planetmath.org/MobiusTransformation";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Mapa de Moebius usando el radio cruzado tomando
infinito a infinito y z2,z3 a 1 y 0 respectivamente.</p><p>Consu
lte <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Mapa de Moebius usando la relación cruzada
tomando de infinito a 1 y z3,z4 a 0 e infinito respectivamente.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Mapa de Moebius usando la relación cruzada
tomando de infinito a 0 y z2,z4 a 1 e infinito respectivamente.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> para o
btener más información.</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>El núcleo de Poisson en D(0,1) (no normalizado a 1, esto es, su integral es
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>El núcleo de Poisson en D(0,R) (no normalizado a
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Alias: <code class="function">zeta</code></p><p>La función «zeta de
Riemann». Actualmente sólo implementada para valores reales.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> o <a class="ulink"
href="http://en.wikipedia.org/wiki/Riemann_zeta_function"; target="_top">Wikipedia</a> para
más información.</p></dd><dt><span class="term"><a
name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre class="synopsis">UnitStep (x)</pre><p>La
función escalón unitario es 0 para x<0, 1 si no. Es la integral de la función delta de Dirac. También
llamada función de Heaviside.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>La función <code class="function">cis</code> es la misma que <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Convertir
grados a radianes.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Convertir
radia
nes a grados.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Calcular la función sinc no normalizada, esto es <strong
class="userinput"><code>sin(x)/x</code></strong>. Si quiere normalizar la función utilice <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>Consulte la <a class="ulink"
href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> para obtener más
información.</p><p>Desde la versión 1.0.16 en adelante.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s11.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Cálculo </td><td
width="20%" align="center"><a a
ccesskey="h" href="index.html">Inicio</a></td><td width="40%" align="right" valign="top"> Resolución de
ecuaciones</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Funciones</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manual
de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de funciones GEL"><link rel="prev"
href="ch11s11.html" title="Cálculo"><link rel="next" href="ch11s13.html" title="Resolución de
ecuaciones"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Funciones</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de funciones
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s13.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-gel-function-list-functions"></a>Funciones</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Alias:
<code class="function">Arg</code><code class="function">arg</code></p><p>argumento (ángulo) de un número
complejo.</p></dd><dt><span class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre
class="synopsis">BesselJ0 (x)</pre><p>Función de Bessel de primer tipo de orden 0. Implementada solo para
números reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Función
de Bessel de primer tipo de orden 1. Implementada solo para números reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Función
de Bessel de primer tipo de orden <code class="varname">n</code>. Implementada solo para números
reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Función
de Bessel de segundo tipo de orden 0. Implementada solo para números reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Función
de Bessel de segunto tipo de orden 1. Implementada solo para números reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Función
de Bessel de segundo tipo de orden <code class="varname">n</code>. Implementada solo para números
reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Núcleo de Dirichlet de orden <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Devuelve 1 si y sólo si todos los elementos son cero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alias: <code class="function">erf</code></p><p>La función de error, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>Consulte la <a class="ulink" href="https://en.wikipedia.org/wiki/Error_function";
target="_top">Wikipedia</a> o <a class="ulink" href="http://planetmath.org/ErrorFunction";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="te
rm"><a name="gel-function-FejerKernel"></a>FejerKernel</span></dt><dd><pre class="synopsis">FejerKernel
(n,t)</pre><p>Núcleo de Fejer de orden <code class="varname">n</code> evaluado en <code
class="varname">t</code></p><p>Consulte <a class="ulink" href="http://planetmath.org/FejerKernel";
target="_top">Planetmath</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alias: <code class="function">Gamma</code></p><p>La función «Gamma». Actualmente sólo
implementada para valores reales.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Devuelve 1 si y sólo si todos los elementos son iguales.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>La rama
principal de la función de Lambert W calculada sólo para los valores reales más grandes o iguales que <strong
class="userinput"><code>-1/e</code></strong>. Es decir, que la función <code class="function">LambertW</code>
es la inversa de la expresión <strong class="userinput"><code>x*e^x</code></strong>. Incluso para una
variable real <code class="varname">x</code> esta expresión no es uno a uno y por lo tanto tiene dos ramas
más <strong class="userinput"><code>[-1/e,0)</code></strong>. Consulte <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> para otras ram
as reales.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>La
rama menos uno «-1» de la función de Lambert W calculada sólo para valores reales más grandes o igual a
<strong class="userinput"><code>-1/e</code></strong> y menor que 0. Es decir, <code
class="function">LambertWm1</code> es la segunda rama de la inversa de <strong
class="userinput"><code>x*e^x</code></strong>. Consulte <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> para la rama
principal.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Buscar el primer valor donde
f(x)=0.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Mapa de Moebius del disco a sí mismo mapeando a 0.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Mapa de Moebius usando el radio cruzado z2,z3,z4 a 1,0 e infinito respectivamente.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Mapa de Moebius usando el radio cruzado tomando
infinito a infinito y z2,z3 a 1 y 0 respectivamente.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Mapa de Moebius usando la relación cruzada
tomando de infinito a 1 y z3,z4 a 0 e infinito respectivamente.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Mapa de Moebius usando la relación cruzada
tomando de infinito a 0 y z2,z4 a 1 e infinito respectivamente.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>El núcleo de Poisson en D(0,1) (no normalizado a 1, esto es, su integral es
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>El núcleo de Poisson en D(0,R) (no normalizado a
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Alias: <code class="function">zeta</code></p><p>La función «zeta de
Riemann». Actualmente sólo implementada para valores reales.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>La función escalón unitario es 0 para x<0, 1 si no. Es la integral
de la función delta de Dirac. También llamada función de Heaviside.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>La función <code class="function">cis</code> es la misma que <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Convertir
grados a radianes.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Convertir
radianes a grados.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Calcular la función sinc no normalizada, esto es <strong
class="userinput"><code>sin(x)/x</code></strong>. Si quiere normalizar la función utilice <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ </p><p>Desde la versión 1.0.16 en adelante.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s11.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Cálculo </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%" align="right"
valign="top"> Resolución de ecuaciones</td></tr></table></div></body></html>
diff --git a/help/es/html/ch11s13.html b/help/es/html/ch11s13.html
index ed52d54..e82c8b4 100644
--- a/help/es/html/ch11s13.html
+++ b/help/es/html/ch11s13.html
@@ -1,4 +1,27 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Resolución de
ecuaciones</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de
funciones GEL"><link rel="prev" href="ch11s12.html" title="Funciones"><link rel="next" href="ch11s14.html"
title="Estadísticas"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Resolución de ecuaciones</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s12.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de funciones
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s14.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><di
v><h2 class="title" style="clear: both"><a name="genius-gel-function-list-equation-solving"></a>Resolución
de ecuaciones</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span
class="term"><a name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre
class="synopsis">CubicFormula (p)</pre><p>Calcular las raíces de un polinomio cúbico (de grado 3) utilizando
la fórmula cúbica. El polinomio se dará como un vector de coeficientes. Esto es <strong
class="userinput"><code>4*x^3 + 2*x + 1</code></strong> que corresponde al vector <strong
class="userinput"><code>[1,2,0,4]</code></strong>. Devuelve un vector columna de tres soluciones. La primera
solución siempre es la real como un cúbico siempre tiene una solución real.</p><p>Consulte <a class="ulink"
href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>, <a class="ulink"
href="http://mathworld.wolfram.com/CubicFormula.html"; target="_top">Mathworld</a>, o <a class="u
link" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>Utilizar el método clásico de Euler para resolver numéricamente y'=f(x,y) de forma
inicial <code class="varname">x0</code>, <code class="varname">y0</code> pasan a <code
class="varname">x1</code> con <code class="varname">n</code> incrementos, devuelve <code
class="varname">y</code> junto con <code class="varname">x1</code>. Excepto que especifique explícitamente
que quiere utilizar el método clásico de Euler, piense en utilizar <a class="link"
href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a> para resolver ODE.</p><p>Los sistemas se pueden
resolver teniendo a <code class="varname">y</code> como un vector (columna) en cualquier parte. Es decir,
<code class="varname">y0</code> puede ser un ve
ctor en cuyo caso <code class="varname">f</code> será un número <code class="varname">x</code> y un vector
del mismo tamaño para el segundo argumento y devolverá un vector del mismo tamaño.</p><p>Consulte <a
class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html"; target="_top">Mathworld</a> o <a
class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>Utilizar el método clásico de Euler para resolver
numéricamente y'=f(x,y) de forma inicial <code class="varname">x0</code>, <code class="varname">y0</code>
pasan a <code class="varname">x1</code> con <code class="varname">n</code> incrementos, devuelve una matriz
de 2 por <strong class="userinput"><code>n+1</code></strong> con los valores <code class="varname">x</code> e
<code class="varname">y</code>.Excepto que quiera utilizar explícitamente el método clásico de Euler,
utilice mejor <a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a> para
resolver ODE. Adecuado para enlazar con <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> o <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Ejemplo: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Resolución de
ecuaciones</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html" title="Capítulo 11. Lista de
funciones GEL"><link rel="prev" href="ch11s12.html" title="Funciones"><link rel="next" href="ch11s14.html"
title="Estadísticas"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Resolución de ecuaciones</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s12.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de funciones
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s14.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><di
v><h2 class="title" style="clear: both"><a name="genius-gel-function-list-equation-solving"></a>Resolución
de ecuaciones</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span
class="term"><a name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre
class="synopsis">CubicFormula (p)</pre><p>Calcular las raíces de un polinomio cúbico (de grado 3) utilizando
la fórmula cúbica. El polinomio se dará como un vector de coeficientes. Esto es <strong
class="userinput"><code>4*x^3 + 2*x + 1</code></strong> que corresponde al vector <strong
class="userinput"><code>[1,2,0,4]</code></strong>. Devuelve un vector columna de tres soluciones. La primera
solución siempre es la real como un cúbico siempre tiene una solución real.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>Utilizar el método clásico de Euler para resolver numéricamente y'=f(x,y) de forma
inicial <code class="varname">x0</code>, <code class="varname">y0</code> pasan a <code
class="varname">x1</code> con <code class="varname">n</code> incrementos, devuelve <code
class="varname">y</code> junto con <code class="varname">x1</code>. Excepto que especifique explícitamente
que quiere utilizar el método clásico de Euler, piense en utilizar <a class="link"
href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a> para resolver ODE.</p><p>Los sistemas se pueden
resolver teniendo a <code class="varname">y</code> como un vector (columna) en cualquier parte. Es decir,
<code class="varname">y0</code> puede ser un vector en cuyo caso <code class="varname">f</code> será un
número <code class="varname">x</code> y un vector del
mismo tamaño para el segundo argumento y devolverá un vector del mismo tamaño.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
+ Use classical Euler's method to numerically solve y'=f(x,y) for
+ initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
+ <code class="varname">x1</code> with <code class="varname">n</code> increments,
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
+ <code class="varname">x</code> and <code class="varname">y</code> values.
+ Unless you explicitly want to use Euler's method, you should really
+ think about using
+ <a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a>
+ for solving ODE.
+ Suitable
+ for plugging into
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
+ </p><p>Ejemplo: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line =
EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential
growth");</code></strong>
</pre><p>Los sistemas se pueden resolver teniendo a <code class="varname">y</code> como un vector (columna)
en cualquier parte. Es decir, <code class="varname">y0</code> puede ser un vector en cuyo caso <code
class="varname">f</code> será un número <code class="varname">x</code> y un vector del mismo tamaño para el
segundo argumento y devolverá un vector del mismo tamaño.</p><p>La salida para un sistema es todavía una
matriz de n por 2 siendo la segunda entrada un vector. Si quiere dibujar la línea, asegúrese de utilizar fila
de vectores, y aplanar la matriz con <a class="link"
href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>, y pulse sobre las columnas de la derecha.
Ejemplo: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
@@ -9,9 +32,44 @@
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","First");</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</code></strong>
-</pre><p>Consulte <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> o <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.10 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Buscar la raíz de una función utilizando el método
de la bisección. <code class="varname">a</code> y <code class="varname">b</code> son los límites iniciales
del intervalo, <strong class="userinput"><code>f(a)</code></strong> y <strong
class="userinput"><code>f(b)</code></strong> deben tener signos opuestos. <code class="varname">TOL</code> es
la tolerancia deseada y <code class="varname">N</code> es el límite del número de iteraciones a ejecutar, 0
indica sin límites. La función devuelve un vector <strong c
lass="userinput"><code>[success,value,iteration]</code></strong>, donde <code class="varname">success</code>
un booleano que indica el éxito, <code class="varname">value</code> es el último valor calculado, e <code
class="varname">iteration</code> es el número de iteraciones realizadas.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Buscar la raíz de una función utilizando el
método de la posición falsa. <code class="varname">a</code> y <code class="varname">b</code> son los valores
iniciales del intervalo, <strong class="userinput"><code>f(a)</code></strong> y <strong
class="userinput"><code>f(b)</code></strong> deben tener signos opuestos. <code class="varname">TOL</code> es
la tolerancia deseada y <code class="varname">N</code> es el límite del número de iteraciones a ejecutar, 0
indica sin límites. La función devuelve un vecto
r <strong class="userinput"><code>[success,value,iteration]</code></strong>, donde <code
class="varname">success</code> es un booleano que indica el éxito, <code class="varname">value</code> es el
último valor calculado, e <code class="varname">iteration</code> es el número de iteraciones
realizadas.</p></dd><dt><span class="term"><a
name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Buscar la raíz de una función utilizando el
método de Muller. <code class="varname">TOL</code> es la tolerancia deseada y <code class="varname">N</code>
es el límite del número de iteraciones a ejecutar, 0 indica sin límites. La función devuelve un vector
<strong class="userinput"><code>[success,value,iteration]</code></strong>, donde <code
class="varname">success</code> un booleano que indica el éxito, <code class="varname">value</code> es el
último valor calculado, e <code class="
varname">iteration</code> es el número de iteraciones realizadas.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Buscar la raíz de una función utilizando el método de la secante. <code
class="varname">a</code> y <code class="varname">b</code> son los límites iniciales del intervalo, <strong
class="userinput"><code>f(a)</code></strong> y <strong class="userinput"><code>f(b)</code></strong> deben
tener signos opuestos. <code class="varname">TOL</code> es la tolerancia deseada y <code
class="varname">N</code> es el límite del número de iteraciones a ejecutar, 0 indica sin límites. La función
devuelve un vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, donde <code
class="varname">success</code> es un booleano que indica el éxito, <code class="varname">value</code> es el
último valor calculado, e <code class="varname">itera
tion</code> es el número de iteraciones realizadas.</p></dd><dt><span class="term"><a
name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,guess,epsilon,maxn)</pre><p>Encontrar ceros utilizando el método de Halleys. Siendo <code
class="varname">f</code> la función, <code class="varname">df</code> es la derivada de <code
class="varname">f</code>, y <code class="varname">ddf</code> es la segunda derivada de <code
class="varname">f</code>. La variable <code class="varname">guess</code> es la aproximación inicial. La
función devuelve después dos valores sucesivos que están dentro de los límites que marca <code
class="varname">epsilon</code> o después de <code class="varname">maxn</code> iteraciones en cuyo caso
devuelve <code class="constant">null</code> indicando un fallo.</p><p>Consulte también <a class="link"
href="ch11s13.html#gel-function-NewtonsMethod"><code class="function">NewtonsMethod</code></a> y <a
class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Ejemplo para encontrar la raíz cuadrada de 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
-</pre><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> para más información.</p><p>Desde la versión 1.0.18 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Encontrar ceros utilizando el método de Newton. La variable <code
class="varname">f</code> es la función y <code class="varname">df</code> es la derivada de <code
class="varname">f</code>. La variable <code class="varname">guess</code> el supuesto inicial. La función
devuelve después dos valores sucesivos que están dentro de los límites que marca <code
class="varname">epsilon</code> o después de <code class="varname">maxn</code> iteraciones en cuyo caso
devuelve <code class="constant">null</code> indicando un fallo.</p><p>Consulte también <a class="link"
href="ch11s15.html#gel-function-NewtonsMethodPoly"><code clas
s="function">NewtonsMethodPoly</code></a> y <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Ejemplo para encontrar la raíz cuadrade de 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
-</pre><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.18 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Calcular las raíces de un polinomio (de grado 1 a 4) utilizando una de las fórmulas para cada
polinomio. El polinomio entregará un vector de coeficientes. Esto es <strong class="userinput"><code>4*x^3 +
2*x + 1</code></strong> que corresponde al vector <strong class="userinput"><code>[1,2,0,4]</code></strong>.
Devuelve un vector columna de las soluciones.</p><p>La función llama a <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a>, y a <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>
.</p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre
class="synopsis">QuadraticFormula (p)</pre><p>Calcular las raíces de una polinomio cuadrático (de grado 2)
utilizando la fórmula cuadrática. El polinomio será un vector de coeficientes. Es es <strong
class="userinput"><code>3*x^2 + 2*x + 1</code></strong> que corresponde con el vector <strong
class="userinput"><code>[1,2,3]</code></strong>. Devuelve un vector columna de las dos
soluciones.</p><p>Consulte <a class="ulink" href="http://planetmath.org/QuadraticFormula";
target="_top">Planetmath</a> o <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Calcular las raíces de un polinomio cuadrático (de grado 4) utilizando la fórm
ula cuadrática. El polinomio será un vector de coeficientes. Esto es <strong class="userinput"><code>5*x^4 +
2*x + 1</code></strong> que corresponde con el vector <strong
class="userinput"><code>[1,2,0,0,5]</code></strong>. Devuelve un vector columna de las cuatro
soluciones.</p><p>Consulte <a class="ulink" href="http://planetmath.org/QuarticFormula";
target="_top">Planetmath</a>, <a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, o <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Utilizar el método clásico no adaptativo de cuarto orden Runge-Kutta para resolver
numéricamente y'=f(x,y) que de forma inicial <code class="varname">x0</code>, <code class="varname">y0</code>
tienden a <code class=
"varname">x1</code> con <code class="varname">n</code> incrementos, devuelve <code class="varname">y</code>
en <code class="varname">x1</code>.</p><p>Los sistemas se pueden resolver teniendo a <code
class="varname">y</code> como un vector (columna) en cualquier parte. Es decir, <code
class="varname">y0</code> puede ser un vector en cuyo caso <code class="varname">f</code> será un número
<code class="varname">x</code> y un vector del mismo tamaño para el segundo argumento y devolverá un vector
del mismo tamaño.</p><p>Consulte <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> o <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>Utilizar el método clásico no adaptativo de cuart
o orden Runge-Kutta para resolver numéricamente y'=f(x,y) que de forma inicial <code
class="varname">x0</code>, <code class="varname">y0</code> tienden a <code class="varname">x1</code> con
<code class="varname">n</code> incrementos, devuelve una matriz de 2 por <strong
class="userinput"><code>n+1</code></strong> con los valores <code class="varname">x</code> e <code
class="varname">y</code>. Adecuado para enlazar con <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> o <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Example: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Desde la versión 1.0.10 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Buscar la raíz de una función utilizando el método
de la bisección. <code class="varname">a</code> y <code class="varname">b</code> son los límites iniciales
del intervalo, <strong class="userinput"><code>f(a)</code></strong> y <strong
class="userinput"><code>f(b)</code></strong> deben tener signos opuestos. <code class="varname">TOL</code> es
la tolerancia deseada y <code class="varname">N</code> es el límite del número de iteraciones a ejecutar, 0
indica sin límites. La función devuelve un vector <strong
class="userinput"><code>[success,value,iteration]</code></strong>, donde <code class="varname">success</code>
un booleano que indica el éxito, <code class="varname">value</code> es el último valor calculado, e <code
class="varname">iteration</code
es el número de iteraciones realizadas.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Buscar la raíz de una función utilizando el
método de la posición falsa. <code class="varname">a</code> y <code class="varname">b</code> son los
valores iniciales del intervalo, <strong class="userinput"><code>f(a)</code></strong> y <strong
class="userinput"><code>f(b)</code></strong> deben tener signos opuestos. <code class="varname">TOL</code>
es la tolerancia deseada y <code class="varname">N</code> es el límite del número de iteraciones a
ejecutar, 0 indica sin límites. La función devuelve un vector <strong
class="userinput"><code>[success,value,iteration]</code></strong>, donde <code
class="varname">success</code> es un booleano que indica el éxito, <code class="varname">value</code> es
el último valor calculado, e <code class="varname">i
teration</code> es el número de iteraciones realizadas.</p></dd><dt><span class="term"><a
name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Buscar la raíz de una función utilizando el
método de Muller. <code class="varname">TOL</code> es la tolerancia deseada y <code class="varname">N</code>
es el límite del número de iteraciones a ejecutar, 0 indica sin límites. La función devuelve un vector
<strong class="userinput"><code>[success,value,iteration]</code></strong>, donde <code
class="varname">success</code> un booleano que indica el éxito, <code class="varname">value</code> es el
último valor calculado, e <code class="varname">iteration</code> es el número de iteraciones
realizadas.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Buscar la raíz de
una función utilizando el método de la secante. <code class="varname">a</code> y <code
class="varname">b</code> son los límites iniciales del intervalo, <strong
class="userinput"><code>f(a)</code></strong> y <strong class="userinput"><code>f(b)</code></strong> deben
tener signos opuestos. <code class="varname">TOL</code> es la tolerancia deseada y <code
class="varname">N</code> es el límite del número de iteraciones a ejecutar, 0 indica sin límites. La función
devuelve un vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, donde <code
class="varname">success</code> es un booleano que indica el éxito, <code class="varname">value</code> es el
último valor calculado, e <code class="varname">iteration</code> es el número de iteraciones
realizadas.</p></dd><dt><span class="term"><a
name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,guess,epsilon,maxn)</pre><p>Encontrar ceros util
izando el método de Halleys. Siendo <code class="varname">f</code> la función, <code
class="varname">df</code> es la derivada de <code class="varname">f</code>, y <code
class="varname">ddf</code> es la segunda derivada de <code class="varname">f</code>. La variable <code
class="varname">guess</code> es la aproximación inicial. La función devuelve después dos valores sucesivos
que están dentro de los límites que marca <code class="varname">epsilon</code> o después de <code
class="varname">maxn</code> iteraciones en cuyo caso devuelve <code class="constant">null</code> indicando un
fallo.</p><p>Consulte también <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code
class="function">NewtonsMethod</code></a> y <a class="link"
href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Ejemplo para encontrar la raíz cuadrada de 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong cla
ss="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Encontrar ceros utilizando el método de Newton. La variable <code
class="varname">f</code> es la función y <code class="varname">df</code> es la derivada de <code
class="varname">f</code>. La variable <code class="varname">guess</code> el supuesto inicial. La función
devuelve después dos valores sucesivos que están dentro de los límites que marca <code
class="varname">epsilon</code> o después de <code class="varname">maxn</code> iteraciones en cuyo caso
devuelve <code class="constant">null</code> indicando un fallo.</p><p>Consulte también <a class="link"
href="ch11s15.html#gel-function-NewtonsMethodPoly"><code class="function">NewtonsMethodPoly</code></a> y <a
class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code class="function">Symbo
licDerivative</code></a>.</p><p>Ejemplo para encontrar la raíz cuadrade de 10: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Calcular las raíces de un polinomio (de grado 1 a 4) utilizando una de las fórmulas para cada
polinomio. El polinomio entregará un vector de coeficientes. Esto es <strong class="userinput"><code>4*x^3 +
2*x + 1</code></strong> que corresponde al vector <strong class="userinput"><code>[1,2,0,4]</code></strong>.
Devuelve un vector columna de las soluciones.</p><p>La función llama a <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a>, y a <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre class="synopsis">QuadraticFormu
la (p)</pre><p>Calcular las raíces de una polinomio cuadrático (de grado 2) utilizando la fórmula
cuadrática. El polinomio será un vector de coeficientes. Es es <strong class="userinput"><code>3*x^2 + 2*x +
1</code></strong> que corresponde con el vector <strong class="userinput"><code>[1,2,3]</code></strong>.
Devuelve un vector columna de las dos soluciones.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Calcular las raíces de un polinomio cuadrático (de grado 4) utilizando la fórmula cuadrática. El
polinomio será un vector de coeficientes. Esto es <strong class="userinput"><code>5*x^4 + 2*x +
1</code></strong> que corresponde con el vector <strong class="userinput"><code>[1,2,0,0,5]</code></strong>.
Devuelve un vector columna de las cuatro soluciones.</p><p>
+ See
+ <a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
+ <a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Utilizar el método clásico no adaptativo de cuarto orden Runge-Kutta para resolver
numéricamente y'=f(x,y) que de forma inicial <code class="varname">x0</code>, <code class="varname">y0</code>
tienden a <code class="varname">x1</code> con <code class="varname">n</code> incrementos, devuelve <code
class="varname">y</code> en <code class="varname">x1</code>.</p><p>Los sistemas se pueden resolver teniendo a
<code class="varname">y</code> como un vector (columna) en cualquier parte. Es decir, <code
class="varname">y0</code> puede ser un vector en cuyo caso <code class="varname">f</code> será un número
<code class="varname">x</code> y un vector del mismo tamaño para el segundo argumento y devolverá un vector
del mismo tamaño.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
+ Use classical non-adaptive fourth order Runge-Kutta method to
+ numerically solve
+ y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
+ going to <code class="varname">x1</code> with <code class="varname">n</code>
+ increments,
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
+ <code class="varname">x</code> and <code class="varname">y</code> values. Suitable
+ for plugging into
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
+ <a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
+ </p><p>Example: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line =
RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential
growth");</code></strong>
</pre><p>Los sistemas se pueden resolver teniendo a <code class="varname">y</code> como un vector (columna)
en cualquier parte. Es decir, <code class="varname">y0</code> puede ser un vector en cuyo caso <code
class="varname">f</code> será un número <code class="varname">x</code> y un vector del mismo tamaño para el
segundo argumento y devolverá un vector del mismo tamaño.</p><p>La salida de un sistema todavía es una matriz
de n por 2 siendo la segunda entrada un vector. Si quiere dibujar la línea, asegúrese de utilizar filas de
vectores, y aplane la matriz con <a class="link"
href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>, y pulse a la derecha de las columnas.
Ejemplo: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
@@ -22,4 +80,8 @@
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","First");</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</code></strong>
-</pre><p>Consulte <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> o <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> para obtener más información.</p><p>Desde la versión 1.0.10 en
adelante.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s12.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s14.html">Siguiente</a></td></tr><tr><td width="40%" align="left"
valign="top">Funciones </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top">
Estadísticas</td></tr></table></div></body></html>
+</pre><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ </p><p>Desde la versión 1.0.10 en adelante.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s12.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Funciones </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%" align="right"
valign="top"> Estadísticas</td></tr></table></div></body></html>
diff --git a/help/es/html/ch11s14.html b/help/es/html/ch11s14.html
index aa7630e..91938c0 100644
--- a/help/es/html/ch11s14.html
+++ b/help/es/html/ch11s14.html
@@ -1 +1,26 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Estadísticas</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s13.html" title="Resolución de
ecuaciones"><link rel="next" href="ch11s15.html" title="Polinomios"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Estadísticas</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class
="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Estadísticas</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alias: <code
class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Calcular la media de una matriz entera.</p><p>Consulte <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral de la función de Gauss desde 0 a <code
class="varname">x</code> (área debajo de la curva normal).</p><p>Consulte <a class="ulink"
href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld<
/a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>La función de distribución Gausiana normalizada (la curva normal).</p><p>Consulte <a
class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Median"></a>Median</span></dt><dd><pre class="synopsis">Median (m)</pre><p>Alias: <code
class="function">median</code></p><p>Calcular la mediana de una matriz entera.</p><p>Consulte <a
class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> para
obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">stdevp</code></p><p>Calcular la desviación de población típica de una matriz
completa.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Alias: <code class="function">RowMean</code></p><p>Calcular la media
de cada columna de una matriz.</p><p>Consulte <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> para obtener más
información.</p></dd><dt><span class="term"><a
name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre class="synopsis">RowMedian
(m)</pre><p>Calcular la mediana de cada fila en una matriz y devolver una vector columna de las
medianas.</p><p>Consulte <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandard
Deviation</span></dt><dd><pre class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">rowstdevp</code></p><p>Calcular la desviación típica de las columnas de una matriz y
devuelve una matriz columna.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alias: <code
class="function">rowstdev</code></p><p>Calcular la desviación estándar de las filas de una matriz y
devuelve una matriz columna.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alias: <code class="function">stdev</code></p><p>Calcular la
desviación estándar de una matriz entera.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s13.htm
l">Anterior</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td
width="40%" align="right"> <a accesskey="n" href="ch11s15.html">Siguiente</a></td></tr><tr><td width="40%"
align="left" valign="top">Resolución de ecuaciones </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top">
Polinomios</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Estadísticas</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s13.html" title="Resolución de
ecuaciones"><link rel="next" href="ch11s15.html" title="Polinomios"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Estadísticas</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class
="title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Estadísticas</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alias: <code
class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral de la función de Gauss desde 0 a <code
class="varname">x</code> (área debajo de la curva normal).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>La función de distribución Gausiana normalizada (la curva normal).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Alias: <code class="function">median</code></p><p>Calcular la mediana de
una matriz entera.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">stdevp</code></p><p>Calcular la desviación de población típica de una matriz
completa.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Alias: <code class="function">RowMean</code></p><p>Calculate average
of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calcular la mediana de cada fila en una matriz y devolver una vector
columna de las medianas.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">rowstdevp</code></p><p>Calcular la desviación típica de las columnas de una matriz y
devuelve una matriz columna.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alias: <code
class="function">rowstdev</code></p><p>Calcular la desviación estándar de las filas de una matriz y
devuelve una matriz columna.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alias: <code class="function">stdev</code></p><p>Calcular la
desviación estándar de una matriz entera.</p></dd></dl></div></div><div class="navfooter"><hr><
table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Resolución de
ecuaciones </td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td
width="40%" align="right" valign="top"> Polinomios</td></tr></table></div></body></html>
diff --git a/help/es/html/ch11s15.html b/help/es/html/ch11s15.html
index 6d7c1db..7bd9ad9 100644
--- a/help/es/html/ch11s15.html
+++ b/help/es/html/ch11s15.html
@@ -1,2 +1,5 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Polinomios</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s14.html" title="Estadísticas"><link
rel="next" href="ch11s16.html" title="Teoría de conjuntos"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Polinomios</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s14.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. Lista de
funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s16.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-function-list-polynomials"></a>Polinomios</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AddPoly"></a>AddPoly</span></dt><dd><pre class="synopsis">AddPoly (p1,p2)</pre><p>Suma dos
polinomios (vectores).</p></dd><dt><span class="term"><a
name="gel-function-DividePoly"></a>DividePoly</span></dt><dd><pre class="synopsis">DividePoly
(p,q,&r)</pre><p>Dividir dos polinomios (como vectores) utilizando la división larga. Devuelve el
cociente de los dos polinomios. El argumento opcional <code class="varname">r</code> se utiliza para devolver
el residuo. El residuo tendrá el grado más bajo que <code class="varname">q</code>.</p><p>Consulte <a
class="ulink" href="http://planetmath.org/PolynomialLongDivision"; target="_top">Planetmath</a> para obtener
más información.</p></dd><dt><span class="term"><a name="gel-function-IsPoly"></a>IsPoly</span></dt><dd><pre
class="syno
psis">IsPoly (p)</pre><p>Comprobar si un vector se puede usar como un polinomio.</p></dd><dt><span
class="term"><a name="gel-function-MultiplyPoly"></a>MultiplyPoly</span></dt><dd><pre
class="synopsis">MultiplyPoly (p1,p2)</pre><p>Multiplica dos polinomios (como vectores).</p></dd><dt><span
class="term"><a name="gel-function-NewtonsMethodPoly"></a>NewtonsMethodPoly</span></dt><dd><pre
class="synopsis">NewtonsMethodPoly (poly,guess,epsilon,maxn)</pre><p>Encontrar una raíz de un polinomio
utilizando el método de Newton. La variable <code class="varname">poly</code> es el polinomio en forma
vectorial y <code class="varname">guess</code> es la suposición inicial. La función devuelve después dos
valores sucesivos que están dentro de los límites que marca <code class="varname">epsilon</code> o después de
<code class="varname">maxn</code> iteraciones en cuyo caso devuelve <code class="constant">null</code>
indicando un fallo.</p><p>Consulte también <a class="link" href="ch
11s13.html#gel-function-NewtonsMethod"><code class="function">NewtonsMethod</code></a>.</p><p>Ejemplo para
encontrar la raíz cuadrada de 10: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethodPoly([-10,0,1],3,10^-10,100)</code></strong>
-</pre><p>Consulte la <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method";
target="_top">Wikipedia</a> para obtener más información.</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Tomar la derivada segunda (como vector)
polinómico.</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Tomar la derivada (como vector) polinómico.</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Extraer una función de un polinomio (como vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Extraer una cadena de un polinomio (como vector).</p></dd><dt><span c
lass="term"><a name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre
class="synopsis">SubtractPoly (p1,p2)</pre><p>Restar dos polinomios (como vectores).</p></dd><dt><span
class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly
(p)</pre><p>Eliminar ceros de un polinomio (como vector).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Estadísticas
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%"
align="right" valign="top"> Teoría de conjuntos</td></tr></table></div></body></html>
+</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Tomar la derivada segunda (como vector)
polinómico.</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Tomar la derivada (como vector) polinómico.</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Extraer una función de un polinomio (como vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Extraer una cadena de un polinomio (como vector).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Restar dos pol
inomios (como vectores).</p></dd><dt><span class="term"><a
name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly (p)</pre><p>Eliminar
ceros de un polinomio (como vector).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s14.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Estadísticas
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%"
align="right" valign="top"> Teoría de conjuntos</td></tr></table></div></body></html>
diff --git a/help/es/html/ch11s18.html b/help/es/html/ch11s18.html
index 5d75550..093052f 100644
--- a/help/es/html/ch11s18.html
+++ b/help/es/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscelánea</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s17.html" title="Álgebra
conmutativa"><link rel="next" href="ch11s19.html" title="Operaciones simbólicas"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Miscelánea</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><th width="60%"
align="center">Capítulo 11. Lista de funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 cla
ss="title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Miscelánea</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convertir un vector de valores ASCII en una cadena.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alfabeto)</pre><p>Convierte un vector de valores alfabéticos basados
en 0 (posiciones en la cadena alfabeto) a una cadena.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(cad)</pre><p>Convertir una cadena a un vector de valores ASCII.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (
str,alfabeto)</pre><p>Convertir una cadena en un vector de valores alfabéticos basados en 0 (posiciones en
la cadena alfabeto), -1 para las letras desconocidas.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s17.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Subir</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top">Álgebra conmutativa
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Inicio</a></td><td width="40%"
align="right" valign="top"> Operaciones simbólicas</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscelánea</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual de Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. Lista de funciones GEL"><link rel="prev" href="ch11s17.html" title="Álgebra
conmutativa"><link rel="next" href="ch11s19.html" title="Operaciones simbólicas"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Miscelánea</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><th width="60%"
align="center">Capítulo 11. Lista de funciones GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Siguiente</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 cla
ss="title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Miscelánea</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alfabeto)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(cad)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alfabeto)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Subir</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Siguiente</a></td></tr><tr><td width="40%" align="left"
valign="top">Álgebra conmutativa </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Inicio</a></td><td width="40%" align="right" valign="top"> Operaciones
simbólicas</td></tr></table></div></body></html>
diff --git a/help/es/html/ch11s20.html b/help/es/html/ch11s20.html
index f4a0ab1..43f9d17 100644
--- a/help/es/html/ch11s20.html
+++ b/help/es/html/ch11s20.html
@@ -2,18 +2,24 @@
<code class="prompt">genius></code> <strong
class="userinput"><code>ExportPlot("/carpeta/archivo","eps")</code></strong>
</pre><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-LinePlot"></a>LinePlot</span></dt><dd><pre class="synopsis">LinePlot
(func1,func2,func3,...)</pre><pre class="synopsis">LinePlot (func1,func2,func3,x1,x2)</pre><pre
class="synopsis">LinePlot (func1,func2,func3,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlot
(func1,func2,func3,[x1,x2])</pre><pre class="synopsis">LinePlot
(func1,func2,func3,[x1,x2,y1,y2])</pre><p>Dibujar una función (o varias funciones) con una línea. Los 10
primeros argumentos son funciones, entonces opcionalmente puede especificar los límites de las gráficas como
<code class="varname">x1</code>, <code class="varname">x2</code>, <code class="varname">y1</code>, <code
class="varname">y2</code>. Si no se especifican los límites, entonces se aplican los límites actuales
(Consulte <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>). Si no se espec
ifican los límites de y, las funciones se calculan y se usan las áreas máxima y mínima.</p><p>El parámetro
<a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> controla el dibujado de la leyenda.</p><p>Ejemplos: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlot(sin,cos)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlot(`(x)=x^2,-1,1,0,1)</code></strong>
-</pre></dd><dt><span class="term"><a name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre
class="synopsis">LinePlotClear ()</pre><p>Muestra la ventana de dibujo lineal y limpia las funciones y otras
líneas que se hubiesen dibujado.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (func,...)</pre><pre class="synopsis">LinePlotCParametric
(func,t1,t2,tinc)</pre><pre class="synopsis">LinePlotCParametric
(func,t1,t2,tinc,x1,x2,y1,y2)</pre><p>Dibujar una función valorada paramétrica compleja con una línea.
Primero vienen las funciones que devuelven <code class="computeroutput">x+iy</code>, luego, opcionalmente,
los <code class="varname">t</code> límites como <strong class="userinput"><code>t1,t2,tinc</code></strong>, y
límites como <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Si los límites no se
especifican, entonces se aplican
las configuraciones actuales (Consulte <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Si en
lugar de la cadena se da el valor «fit» para los límites x e y, los límites son la medida máxima de la
gráfica.</p><p>El parámetro <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> controla el dibujado de la leyenda.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>Dibuja una línea desde <code class="varname">x1</code>,<code class="varname">y1</code> a
<code class="varname">x2</code>,<code class="varname">y2</code>. Es posible reemplazar <code
class="varname">x1</code>,<code class="varname">y1</code>, <code class="varname">x2</code>,<code
class="varname">y2</code> p
or una matriz de <code class="varname">n</code> por 2 para obtener una curva poligonal de mayor longitud.
También el vector <code class="varname">v</code> puede ser un vector columna de números complejos, esto es
una matriz <code class="varname">n</code> por 1 y cada número complejo se considera un punto en el
plano.</p><p>Se pueden añadir parámetros adicionales para especificar el color de la línea, ancho, flechas,
ventanas de dibujado o leyendas. Puede modificarlo añadiendo un valor a <strong
class="userinput"><code>«color»</code></strong>, <strong class="userinput"><code>«ancho»</code></strong>,
<strong class="userinput"><code>«ventana»</code></strong>, <strong
class="userinput"><code>«flecha»</code></strong>, o <strong
class="userinput"><code>«leyenda»</code></strong>, y después especificar su color, la anchura, la ventana
como 4 vectores, tipo de flecha, o la leyenda. (Flecha y ventana están desde la versión 1.0.6 y
posteriores.)</p><p>Si la línea s
e considera como un polígono relleno, relleno con el color dado, se puede especificar el argumento <strong
class="userinput"><code>«llenado»</code></strong>. Desde la versión 1.0.22 en adelante.</p><p>La denominación
del color debe ser una cadena que identifique al color según el diccionario inglés que GTK reconocerá como
<strong class="userinput"><code>«red»</code></strong>, <strong
class="userinput"><code>«blue»</code></strong>, <strong class="userinput"><code>«yellow»</code></strong>,
etc... De forma alternativa el color se puede especificar en formato RGB como por ejemplo <strong
class="userinput"><code>«#rgb»</code></strong>, <strong class="userinput"><code>«#rrggbb»</code></strong>, o
<strong class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de
los colores rojo, verde y azul (red, green, blue) . Finalmente, desde la versión 1.0.18, los colores se
pueden especificar como vectores siendo el rojo, verd
e y azul componentes con valores que solo pueden ser 0 o 1. Por ejemplo: <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Los valores de entrada de la ventana deben ser
del tipo <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, o bien, pueden ser una cadena <strong
class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango de x se establecerá con
precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de la línea.</p><p>La
especificación para la flecha debería ser <strong class="userinput"><code>«origen»</code></strong>, <strong
class="userinput"><code>«fin»</code></strong>, <strong class="userinput"><code>«ambos»</code></strong>, o
<strong class="userinput"><code>«ninguno»</code></strong>.</p><p>Finalmente, la leyenda debería ser una
cadena que se pueda utilizar como leyenda en un gráfico. Es decir, si se imprimen las
leyendas.</p><p>Ejemplos: </p><pre class="screen"><code class="p
rompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
+</pre></dd><dt><span class="term"><a name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre
class="synopsis">LinePlotClear ()</pre><p>Muestra la ventana de dibujo lineal y limpia las funciones y otras
líneas que se hubiesen dibujado.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (func,...)</pre><pre class="synopsis">LinePlotCParametric
(func,t1,t2,tinc)</pre><pre class="synopsis">LinePlotCParametric
(func,t1,t2,tinc,x1,x2,y1,y2)</pre><p>Dibujar una función valorada paramétrica compleja con una línea.
Primero vienen las funciones que devuelven <code class="computeroutput">x+iy</code>, luego, opcionalmente,
los <code class="varname">t</code> límites como <strong class="userinput"><code>t1,t2,tinc</code></strong>, y
límites como <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Si los límites no se
especifican, entonces se aplican
las configuraciones actuales (Consulte <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Si en
lugar de la cadena se da el valor «fit» para los límites x e y, los límites son la medida máxima de la
gráfica.</p><p>El parámetro <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> controla el dibujado de la leyenda.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>Dibuja una línea desde <code class="varname">x1</code>,<code class="varname">y1</code> a
<code class="varname">x2</code>,<code class="varname">y2</code>. Es posible reemplazar <code
class="varname">x1</code>,<code class="varname">y1</code>, <code class="varname">x2</code>,<code
class="varname">y2</code> p
or una matriz de <code class="varname">n</code> por 2 para obtener una curva poligonal de mayor longitud.
También el vector <code class="varname">v</code> puede ser un vector columna de números complejos, esto es
una matriz <code class="varname">n</code> por 1 y cada número complejo se considera un punto en el
plano.</p><p>Se pueden añadir parámetros adicionales para especificar el color de la línea, ancho, flechas,
ventanas de dibujado o leyendas. Puede modificarlo añadiendo un valor a <strong
class="userinput"><code>«color»</code></strong>, <strong class="userinput"><code>«ancho»</code></strong>,
<strong class="userinput"><code>«ventana»</code></strong>, <strong
class="userinput"><code>«flecha»</code></strong>, o <strong
class="userinput"><code>«leyenda»</code></strong>, y después especificar su color, la anchura, la ventana
como 4 vectores, tipo de flecha, o la leyenda. (Flecha y ventana están desde la versión 1.0.6 y
posteriores.)</p><p>Si la línea s
e considera como un polígono relleno, relleno con el color dado, se puede especificar el argumento <strong
class="userinput"><code>«llenado»</code></strong>. Desde la versión 1.0.22 en adelante.</p><p>La denominación
del color debe ser una cadena que identifique al color según el diccionario inglés que GTK reconocerá como
<strong class="userinput"><code>«red»</code></strong>, <strong
class="userinput"><code>«blue»</code></strong>, <strong class="userinput"><code>«yellow»</code></strong>,
etc... De forma alternativa el color se puede especificar en formato RGB como por ejemplo <strong
class="userinput"><code>«#rgb»</code></strong>, <strong class="userinput"><code>«#rrggbb»</code></strong>, o
<strong class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de
los colores rojo, verde y azul (red, green, blue) . Finalmente, desde la versión 1.0.18, los colores se
pueden especificar como vectores siendo el rojo, verd
e y azul componentes con valores que solo pueden ser 0 o 1. Por ejemplo: <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Los valores de entrada de la ventana deben ser
del tipo <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, o bien, pueden ser una cadena <strong
class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango de x se establecerá con
precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de la línea.</p><p>La
especificación para la flecha debería ser <strong class="userinput"><code>«origen»</code></strong>, <strong
class="userinput"><code>«fin»</code></strong>, <strong class="userinput"><code>«ambos»</code></strong>, o
<strong class="userinput"><code>«ninguno»</code></strong>.</p><p>Finalmente, la leyenda debería ser una
cadena que se pueda utilizar como leyenda en un gráfico. Es decir, si se imprimen las leyendas.</p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
-</pre><p>A diferencia de muchas otras funciones que no les importa si toman una columna o un vector fila, si
se especifican puntos como un vector de valores complejos, debido a las posibles ambigüedades, es preferible
que sea un vector columna.</p><p>La especificación de <code class="varname">v</code> como un vector columna
de números complejos, se implementa desde la versión 1.0.22 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints
(v,...)</pre><p>Dibuja un punto en <code class="varname">x</code>,<code class="varname">y</code>. La entrada
puede ser una matriz <code class="varname">n</code> por 2 para <code class="varname">n</code> puntos
diferentes. Esta función es esencialmente la misma entrada que <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>. De forma altern
ativa, el vector <code class="varname">v</code> puede ser un vector columna de números complejos, esto es
una matriz <code class="varname">n</code> por 1 y cada número complejo se considera un punto en el
plano.</p><p>Se pueden añadir parámetros adicionales para especificar el color, ancho, ventanas de dibujado o
leyendas. Puede modificarlo añadiendo la palabra <strong class="userinput"><code>«color»</code></strong>,
<strong class="userinput"><code>«ancho»</code></strong>, <strong
class="userinput"><code>«ventana»</code></strong>, o <strong
class="userinput"><code>«leyenda»</code></strong>, y después especificar su color, la anchura, la ventana
como 4 vectores, o la leyenda.</p><p>La denominación del color debe ser una cadena que identifique al color
según el diccionario inglés que GTK reconocerá como <strong class="userinput"><code>«red»</code></strong>,
<strong class="userinput"><code>«blue»</code></strong>, <strong class="userinput"><code>«yellow»<
/code></strong>, etc... De forma alternativa el color se puede especificar en formato RGB como por ejemplo
<strong class="userinput"><code>«#rgb»</code></strong>, <strong
class="userinput"><code>«#rrggbb»</code></strong>, o <strong
class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de los
colores rojo, verde y azul (red, green, blue) . Finalmente los colores se pueden especificar como vectores
siendo el rojo, verde y azul componentes con valores que solo pueden ser 0 o 1.</p><p>Los valores de entrada
de la ventana deben ser del tipo <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, o bien,
pueden ser una cadena <strong class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango
de x se establecerá con precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de
la línea.</p><p>Finalmente, la leyenda debería ser una cadena que se pueda utilizar como leyenda en un
gráfico. Es decir, si se imprimen las leyendas.</p><p>Ejemplos: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
+</pre><p>
+ </p><p>A diferencia de muchas otras funciones que no les importa si toman una columna o un vector
fila, si se especifican puntos como un vector de valores complejos, debido a las posibles ambigüedades, es
preferible que sea un vector columna.</p><p>La especificación de <code class="varname">v</code> como un
vector columna de números complejos, se implementa desde la versión 1.0.22 en adelante.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints
(v,...)</pre><p>Dibuja un punto en <code class="varname">x</code>,<code class="varname">y</code>. La entrada
puede ser una matriz <code class="varname">n</code> por 2 para <code class="varname">n</code> puntos
diferentes. Esta función es esencialmente la misma entrada que <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>. De form
a alternativa, el vector <code class="varname">v</code> puede ser un vector columna de números complejos,
esto es una matriz <code class="varname">n</code> por 1 y cada número complejo se considera un punto en el
plano.</p><p>Se pueden añadir parámetros adicionales para especificar el color, ancho, ventanas de dibujado o
leyendas. Puede modificarlo añadiendo la palabra <strong class="userinput"><code>«color»</code></strong>,
<strong class="userinput"><code>«ancho»</code></strong>, <strong
class="userinput"><code>«ventana»</code></strong>, o <strong
class="userinput"><code>«leyenda»</code></strong>, y después especificar su color, la anchura, la ventana
como 4 vectores, o la leyenda.</p><p>La denominación del color debe ser una cadena que identifique al color
según el diccionario inglés que GTK reconocerá como <strong class="userinput"><code>«red»</code></strong>,
<strong class="userinput"><code>«blue»</code></strong>, <strong class="userinput"><code>«y
ellow»</code></strong>, etc... De forma alternativa el color se puede especificar en formato RGB como por
ejemplo <strong class="userinput"><code>«#rgb»</code></strong>, <strong
class="userinput"><code>«#rrggbb»</code></strong>, o <strong
class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de los
colores rojo, verde y azul (red, green, blue) . Finalmente los colores se pueden especificar como vectores
siendo el rojo, verde y azul componentes con valores que solo pueden ser 0 o 1.</p><p>Los valores de entrada
de la ventana deben ser del tipo <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, o bien,
pueden ser una cadena <strong class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango
de x se establecerá con precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de
la línea.</p><p>Finalmente, la leyenda debería ser una cadena que se pueda utilizar como leyen
da en un gráfico. Es decir, si se imprimen las leyendas.</p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
-</pre><p>A diferencia de muchas otras funciones que no les importa si toman una columna o un vector fila, si
se especifica los puntos como un vector de valores complejos, debido a las posibles ambigüedades, siempre
debe ser suministrado como un vector columna. Por lo tanto, la notificación en el último ejemplo la
transpuesta del vector <strong class="userinput"><code> 0: 6 userinput> para convertirlo en un vector
columna.</code></strong></p><p>Disponible desde la versión 1.0.18 en adelante. La especificación de <code
class="varname">v</code> como un vector columna de números complejos, se implementa desde la versión 1.0.22
en adelante.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotMouseLocation"></a>LinePlotMouseLocation</span></dt><dd><pre
class="synopsis">LinePlotMouseLocation ()</pre><p>Devuelve un vector fila de un punto de la línea de la
pantalla de dibujado correspondiente a la ubicación actual del ratón. Si la trama de línea no es visible,
entonces imprime un error y devuelve <code class="constant"> null constant>. En este caso se debe
ejecutar LinePlot o LinePlotClear LinePlotClear </code></p></dd><dt><span class="term"><a
name="gel-function-LinePlotParametric"></a>LinePlotParametric</span></dt><dd><pre
class="synopsis">LinePlotParametric (xfunc,yfunc,...)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,[x1,x2,y1,y2])</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,"fit")</pre><p>Dibujar una función paramétrica con una línea. Primero vienen las
funciones para <code class="varname">x</code> e <code class="varname">y</code> luego opcionalmente los <code
class="varname">t</code> límites como <strong class="userinput"><code>t1,t2,tinc</code></strong>, y luego,
opcionalmente, los límites como <strong class="use
rinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Si no se especifican los límites x e y, entonces se aplican
las configuraciones actuales (Consulte <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>). Si en lugar de la cadena se da el valor «fit» para los límites x
e y, los límites son la medida máxima de la gráfica.</p><p>El parámetro <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
controla el dibujado de la leyenda.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotWaitForClick"></a>LinePlotWaitForClick</span></dt><dd><pre
class="synopsis">LinePlotWaitForClick ()</pre><p>Si está en el modo de dibujado de lineas, espera por un clic
en la ventana de dibujado de lineas y devuelve la ubicación del clic como un vector fila. Si se cierra la
ventana de la función devuelve inmediatamente con <code class="constant">null</code
. Si la ventana no está en modo de dibujado de lineas, esta se pone de forma automática. Consulte también
<a class="link" href="ch11s20.html#gel-function-LinePlotMouseLocation"><code
class="function">LinePlotMouseLocation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasFreeze"></a>PlotCanvasFreeze</span></dt><dd><pre
class="synopsis">PlotCanvasFreeze ()</pre><p>Congela el dibujo en el lienzo de dibujado de forma temporal.
Esto es útil si necesita dibujar un grupo de elementos y quiere demorar el dibujado para no permitir el
parpadeo de una animación. Después de terminar con el dibujo debería descongelar el lienzo de dibujado
llamando a la función <a class="link" href="ch11s20.html#gel-function-PlotCanvasThaw"><code
class="function">PlotCanvasThaw</code></a>.</p><p>El lienzo está siempre desbloqueado hasta el final de
cualquier proceso, así que nunca permanece bloqueado. El momento en que se muestra una nueva línea de
comandos, por ejemp
lo, el lienzo de dibujado se descongela automáticamente. También tenga en cuenta que las llamadas a congelar
y descongelar puede anidarse de manera segura.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span
class="term"><a name="gel-function-PlotCanvasThaw"></a>PlotCanvasThaw</span></dt><dd><pre
class="synopsis">PlotCanvasThaw ()</pre><p>Descongela el lienzo de dibujado congelado por la función <a
class="link" href="ch11s20.html#gel-function-PlotCanvasFreeze"><code
class="function">PlotCanvasFreeze</code></a> y volver a dibujar el lienzo inmediatamente. El lienzo también
se descongelará al finalizar la ejecución de cualquier programa.</p><p>Desde la versión 1.0.18 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-PlotWindowPresent"></a>PlotWindowPresent</span></dt><dd><pre
class="synopsis">PlotWindowPresent ()</pre><p>Muestra y eleva la ventana de dibujo, creándola si es
necesario. Normalmente, la ventana se crea cuando se invoca a una de l
as funciones de dibujo, pero no siempre la eleva si está debajo de otra ventana. Esta función es buena para
utilizar en un archivo de órdenes llamado «script» en inglés, donde la ventana de dibujo ha sido creada
anteriormente, y por ahora, oculta detrás de la consola u otras ventanas.</p><p>Desde la versión 1.0.19 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldClearSolutions"></a>SlopefieldClearSolutions</span></dt><dd><pre
class="synopsis">SlopefieldClearSolutions ()</pre><p>Borra las soluciones elaboradas por la función <a
class="link" href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldDrawSolution"></a>SlopefieldDrawSolution</span></dt><dd><pre
class="synopsis">SlopefieldDrawSolution (x, y, dx)</pre><p>Cuando un campo de dibujo de gráficas está activo,
dibuja una solución con las condiciones iniciales especi
ficas. El método estándar de Runge-Kutta se usa con incremento <code class="varname">dx</code>. Las
soluciones permanecen en la gráfica hasta que se muestre un dibujo diferente o se llame a <a class="link"
href="ch11s20.html#gel-function-SlopefieldClearSolutions"><code
class="function">SlopefieldClearSolutions</code></a>. También puede utilizar la interfaz gráfica para dibujar
soluciones y especificar las condiciones iniciales con el ratón.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldPlot"></a>SlopefieldPlot</span></dt><dd><pre class="synopsis">SlopefieldPlot
(func)</pre><pre class="synopsis">SlopefieldPlot (func,x1,x2,y1,y2)</pre><p>Dibujar un campo inclinado. La
función <code class="varname">func</code> tomará dos números reales <code class="varname">x</code> e <code
class="varname">y</code>, o un número complejo. De manera opcional se especificarán los límites de la ventana
de dibujo con <code class="varname">x1</code>, <code class="varname"
x2</code>, <code class="varname">y1</code>, <code class="varname">y2</code>. Si no se especifica ningún
límite, se aplicarán los que estén configurados actualmente (Consulte <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>).</p><p>El
parámetro <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> controla el dibujado de la leyenda.</p><p>Ejemplos:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</code></strong>
+</pre><p>
+ </p><p>A diferencia de muchas otras funciones que no les importa si toman una columna o un vector
fila, si se especifica los puntos como un vector de valores complejos, debido a las posibles ambigüedades,
siempre debe ser suministrado como un vector columna. Por lo tanto, la notificación en el último ejemplo la
transpuesta del vector <strong class="userinput"><code> 0: 6 userinput> para convertirlo en un vector
columna.</code></strong></p><p>Disponible desde la versión 1.0.18 en adelante. La especificación de <code
class="varname">v</code> como un vector columna de números complejos, se implementa desde la versión 1.0.22
en adelante.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotMouseLocation"></a>LinePlotMouseLocation</span></dt><dd><pre
class="synopsis">LinePlotMouseLocation ()</pre><p>Devuelve un vector fila de un punto de la línea de la
pantalla de dibujado correspondiente a la ubicación actual del ratón. Si la trama de línea no es
visible, entonces imprime un error y devuelve <code class="constant"> null constant>. En este caso se
debe ejecutar LinePlot o LinePlotClear LinePlotClear </code></p></dd><dt><span class="term"><a
name="gel-function-LinePlotParametric"></a>LinePlotParametric</span></dt><dd><pre
class="synopsis">LinePlotParametric (xfunc,yfunc,...)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,[x1,x2,y1,y2])</pre><pre class="synopsis">LinePlotParametric
(xfunc,yfunc,t1,t2,tinc,"fit")</pre><p>Dibujar una función paramétrica con una línea. Primero vienen las
funciones para <code class="varname">x</code> e <code class="varname">y</code> luego opcionalmente los <code
class="varname">t</code> límites como <strong class="userinput"><code>t1,t2,tinc</code></strong>, y luego,
opcionalmente, los límites como <strong cl
ass="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Si no se especifican los límites x e y, entonces se
aplican las configuraciones actuales (Consulte <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Si en
lugar de la cadena se da el valor «fit» para los límites x e y, los límites son la medida máxima de la
gráfica.</p><p>El parámetro <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> controla el dibujado de la leyenda.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotWaitForClick"></a>LinePlotWaitForClick</span></dt><dd><pre
class="synopsis">LinePlotWaitForClick ()</pre><p>Si está en el modo de dibujado de lineas, espera por un clic
en la ventana de dibujado de lineas y devuelve la ubicación del clic como un vector fila. Si se cierra la
ventana de la función devuelve inmediatamente con <code class="constant">nu
ll</code>. Si la ventana no está en modo de dibujado de lineas, esta se pone de forma automática. Consulte
también <a class="link" href="ch11s20.html#gel-function-LinePlotMouseLocation"><code
class="function">LinePlotMouseLocation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasFreeze"></a>PlotCanvasFreeze</span></dt><dd><pre
class="synopsis">PlotCanvasFreeze ()</pre><p>Congela el dibujo en el lienzo de dibujado de forma temporal.
Esto es útil si necesita dibujar un grupo de elementos y quiere demorar el dibujado para no permitir el
parpadeo de una animación. Después de terminar con el dibujo debería descongelar el lienzo de dibujado
llamando a la función <a class="link" href="ch11s20.html#gel-function-PlotCanvasThaw"><code
class="function">PlotCanvasThaw</code></a>.</p><p>El lienzo está siempre desbloqueado hasta el final de
cualquier proceso, así que nunca permanece bloqueado. El momento en que se muestra una nueva línea de
comandos, p
or ejemplo, el lienzo de dibujado se descongela automáticamente. También tenga en cuenta que las llamadas a
congelar y descongelar puede anidarse de manera segura.</p><p>Desde la versión 1.0.18 en
adelante.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasThaw"></a>PlotCanvasThaw</span></dt><dd><pre class="synopsis">PlotCanvasThaw
()</pre><p>Descongela el lienzo de dibujado congelado por la función <a class="link"
href="ch11s20.html#gel-function-PlotCanvasFreeze"><code class="function">PlotCanvasFreeze</code></a> y volver
a dibujar el lienzo inmediatamente. El lienzo también se descongelará al finalizar la ejecución de cualquier
programa.</p><p>Desde la versión 1.0.18 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-PlotWindowPresent"></a>PlotWindowPresent</span></dt><dd><pre
class="synopsis">PlotWindowPresent ()</pre><p>Muestra y eleva la ventana de dibujo, creándola si es
necesario. Normalmente, la ventana se crea cuando se invoca a
una de las funciones de dibujo, pero no siempre la eleva si está debajo de otra ventana. Esta función es
buena para utilizar en un archivo de órdenes llamado «script» en inglés, donde la ventana de dibujo ha sido
creada anteriormente, y por ahora, oculta detrás de la consola u otras ventanas.</p><p>Desde la versión
1.0.19 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldClearSolutions"></a>SlopefieldClearSolutions</span></dt><dd><pre
class="synopsis">SlopefieldClearSolutions ()</pre><p>Borra las soluciones elaboradas por la función <a
class="link" href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldDrawSolution"></a>SlopefieldDrawSolution</span></dt><dd><pre
class="synopsis">SlopefieldDrawSolution (x, y, dx)</pre><p>Cuando un campo de dibujo de gráficas está activo,
dibuja una solución con las condiciones iniciale
s especificas. El método estándar de Runge-Kutta se usa con incremento <code class="varname">dx</code>. Las
soluciones permanecen en la gráfica hasta que se muestre un dibujo diferente o se llame a <a class="link"
href="ch11s20.html#gel-function-SlopefieldClearSolutions"><code
class="function">SlopefieldClearSolutions</code></a>. También puede utilizar la interfaz gráfica para dibujar
soluciones y especificar las condiciones iniciales con el ratón.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldPlot"></a>SlopefieldPlot</span></dt><dd><pre class="synopsis">SlopefieldPlot
(func)</pre><pre class="synopsis">SlopefieldPlot (func,x1,x2,y1,y2)</pre><p>Dibujar un campo inclinado. La
función <code class="varname">func</code> tomará dos números reales <code class="varname">x</code> e <code
class="varname">y</code>, o un número complejo. De manera opcional se especificarán los límites de la ventana
de dibujo con <code class="varname">x1</code>, <code class="
varname">x2</code>, <code class="varname">y1</code>, <code class="varname">y2</code>. Si no se especifica
ningún límite, se aplicarán los que estén configurados actualmente (Consulte <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>).</p><p>El
parámetro <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> controla el dibujado de la leyenda.</p><p>Ejemplos: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</code></strong>
</pre></dd><dt><span class="term"><a name="gel-function-SurfacePlot"></a>SurfacePlot</span></dt><dd><pre
class="synopsis">SurfacePlot (func)</pre><pre class="synopsis">SurfacePlot (func,x1,x2,y1,y2,z1,z2)</pre><pre
class="synopsis">SurfacePlot (func,x1,x2,y1,y2)</pre><pre class="synopsis">SurfacePlot
(func,[x1,x2,y1,y2,z1,z2])</pre><pre class="synopsis">SurfacePlot (func,[x1,x2,y1,y2])</pre><p>Dibujar una
función superficial que tome entre dos argumentos o un número complejo. Primero vienen las funciones que las
limitan de forma opcional <code class="varname">x1</code>, <code class="varname">x2</code>, <code
class="varname">y1</code>, <code class="varname">y2</code>, <code class="varname">z1</code>, <code
class="varname">z2</code>. Si no se especifican los límites, entonces las configuraciones actuales se
aplicarán (Consulte <a class="link" href="ch11s03.html#gel-function-SurfacePlotWindow"><code
class="function">SurfacePlotWindow</code></a>). Genius sólo puede dibujar
una función superficial sencilla por el momento.</p><p>Si no se especifican los límites de z, se usan los
valores máximo y mínimo de la función.</p><p>Ejemplos: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(|sin|,-1,1,-1,1,0,1.5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(`(x,y)=x^2+y,-1,1,-1,1,-2,2)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlot(`(z)=|z|^2,-1,1,-1,1,0,2)</code></strong>
@@ -27,7 +33,7 @@
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(data,[-1,1,-1,1],"My data")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>d:=null; for i=1 to 20 do for j=1 to
10 do d@(i,j) = (0.1*i-1)^2-(0.1*j)^2;</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(d,[-1,1,0,1],"half a saddle")</code></strong>
-</pre><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>Dibuja una línea desde <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> hasta <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code>. <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code>, <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code> se puede reemplazar por una matriz de <code class="varname">n</code> por 3 para
obtener una curva poligonal de mayor longitud.</p><p>Se pueden añadir parámetros adicionales para especificar
el color de la línea, ancho, flechas, ventanas de dibujado o leyendas. Puede modificarlo a�
�adiendo un valor a <strong class="userinput"><code>«color»</code></strong>, <strong
class="userinput"><code>«ancho»</code></strong>, <strong class="userinput"><code>«ventana»</code></strong>, o
<strong class="userinput"><code>«leyenda»</code></strong>, y después especificar su color, la anchura, la
ventana como 6 vectores, o la leyenda.</p><p>La denominación del color debe ser una cadena que identifique al
color según el diccionario inglés que GTK reconocerá como <strong
class="userinput"><code>«red»</code></strong>, <strong class="userinput"><code>«blue»</code></strong>,
<strong class="userinput"><code>«yellow»</code></strong>, etc... De forma alternativa el color se puede
especificar en formato RGB como por ejemplo <strong class="userinput"><code>«#rgb»</code></strong>, <strong
class="userinput"><code>«#rrggbb»</code></strong>, o <strong
class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de los colo
res rojo, verde y azul (red, green, blue) . Finalmente, desde la versión 1.0.18, los colores se pueden
especificar como vectores siendo el rojo, verde y azul componentes con valores que solo pueden ser 0 o 1. Por
ejemplo: <strong class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Los valores de entrada de la
ventana deben ser del tipo <strong class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, o bien,
pueden ser una cadena <strong class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango
de x se establecerá con precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de
la línea.</p><p>Finalmente, la leyenda debería ser una cadena que se pueda utilizar como leyenda en un
gráfico. Es decir, si se imprimen las leyendas.</p><p>Ejemplos: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
+</pre><p>Desde la versión 1.0.16 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>Dibuja una línea desde <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> hasta <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code>. <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code>, <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code> se puede reemplazar por una matriz de <code class="varname">n</code> por 3 para
obtener una curva poligonal de mayor longitud.</p><p>Se pueden añadir parámetros adicionales para especificar
el color de la línea, ancho, ventanas de dibujado o leyendas. Puede modificarlo añadiendo
un valor a <strong class="userinput"><code>«color»</code></strong>, <strong
class="userinput"><code>«ancho»</code></strong>, <strong class="userinput"><code>«ventana»</code></strong>,o
<strong class="userinput"><code>«leyenda»</code></strong>, y después especificar su color, la anchura, la
ventana como 6 vectores, o la leyenda.</p><p>La denominación del color debe ser una cadena que identifique al
color según el diccionario inglés que GTK reconocerá como <strong
class="userinput"><code>«red»</code></strong>, <strong class="userinput"><code>«blue»</code></strong>,
<strong class="userinput"><code>«yellow»</code></strong>, etc... De forma alternativa el color se puede
especificar en formato RGB como por ejemplo <strong class="userinput"><code>«#rgb»</code></strong>, <strong
class="userinput"><code>«#rrggbb»</code></strong>, o <strong
class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de los
colores rojo,
verde y azul (red, green, blue) . Finalmente, desde la versión 1.0.18, los colores se pueden especificar
como vectores siendo el rojo, verde y azul componentes con valores que solo pueden ser 0 o 1. Por ejemplo:
<strong class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Los valores de entrada de la ventana
deben ser del tipo <strong class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, o bien, pueden ser
una cadena <strong class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango de x se
establecerá con precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de la
línea.</p><p>Finalmente, la leyenda debería ser una cadena que se pueda utilizar como leyenda en un gráfico.
Es decir, si se imprimen las leyendas.</p><p>Ejemplos: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine([0,0,0;1,-1,2;-1,-1,-3])</code></strong>
</pre><p>Disponible desde la versión 1.0.19 en adelante.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawPoints"></a>SurfacePlotDrawPoints</span></dt><dd><pre
class="synopsis">SurfacePlotDrawPoints (x,y,z,...)</pre><pre class="synopsis">SurfacePlotDrawPoints
(v,...)</pre><p>Dibuja un punto en <code class="varname">x</code>,<code class="varname">y</code>,<code
class="varname">z</code>. La entrada puede ser una <code class="varname">n</code> por 3 matriz para <code
class="varname">n</code> puntos diferentes. Esta función es esencialmente la misma entrada que <a
class="link" href="ch11s20.html#gel-function-SurfacePlotDrawLine">SurfacePlotDrawLine</a>.</p><p>Se pueden
añadir parámetros adicionales para especificar el color de la línea, ancho, ventanas de dibujado o leyendas.
Puede modificarlo añadiendo un valor a <strong class="userinput"><code>«color»</code></strong>, <strong
class="userinput"><code>«ancho»</code></strong>, <strong class="userinput
"><code>«ventana»</code></strong>,o <strong class="userinput"><code>«leyenda»</code></strong>, y después
especificar su color, la anchura, la ventana como 6 vectores, o la leyenda.</p><p>La denominación del color
debe ser una cadena que identifique al color según el diccionario inglés que GTK reconocerá como <strong
class="userinput"><code>«red»</code></strong>, <strong class="userinput"><code>«blue»</code></strong>,
<strong class="userinput"><code>«yellow»</code></strong>, etc... De forma alternativa el color se puede
especificar en formato RGB como por ejemplo <strong class="userinput"><code>«#rgb»</code></strong>, <strong
class="userinput"><code>«#rrggbb»</code></strong>, o <strong
class="userinput"><code>«#rrrrggggbbbb»</code></strong>, donde r, g, o b son dígitos hexadecimales de los
colores rojo, verde y azul (red, green, blue) . Finalmente los colores se pueden especificar como vectores
siendo el rojo, verde y azul componentes con valores que sol
o pueden ser 0 o 1.</p><p>Los valores de entrada de la ventana deben ser del tipo <strong
class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, o bien, pueden ser una cadena <strong
class="userinput"><code>«ajuste»</code></strong>, en cualquier caso, el rango de x se establecerá con
precisión y el rango y se puede ajustar con cinco por ciento alrededor del borde de la
línea.</p><p>Finalmente, la leyenda debería ser una cadena que se pueda utilizar como leyenda en un gráfico.
Es decir, si se imprimen las leyendas.</p><p>Ejemplos: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints([0,0,0;1,-1,2;-1,-1,1])</code></strong>
diff --git a/help/es/html/index.html b/help/es/html/index.html
index 6e5df44..7ec27df 100644
--- a/help/es/html/index.html
+++ b/help/es/html/index.html
@@ -1,3 +1,3 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Manual de
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Manual de la herramienta matemática Genius."><link rel="home" href="index.html" title="Manual de
Genius"><link rel="next" href="ch01.html" title="Capítulo 1. Introducción"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Manual de Genius</th></tr><tr><td width="20%"
align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Siguiente</a></td></tr></table><hr></div><div lang="es" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Manual de Genius</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><sp
an class="firstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Universidad del estado de Oklahoma<br></span><div class="address"><p> <code
class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code>
</p></div></div></div><div class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">Universidad de
Queensland, Australia<br></span><div class="address"><p> <code class="email"><<a class="email"
href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">Este manual describe la versión 1.0.22 de
Genius.</p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2011 Daniel
Mustieles (d
aniel mustieles gmail com)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Se concede
permiso para copiar, distribuir o modificar este documento según las condiciones de la GNU Free
Documentation License (GFDL), Versión 1.1 o cualquier versión posterior publicada por la Free Software
Foundation sin Secciones invariantes, Textos de portada y Textos de contraportada. Encontrará una copia de
la GFDL en este <a class="ulink" href="ghelp:fdl" target="_top">enlace</a> o en el archivo COPYING-DOCS
distribuido con este manual.</p><p>Este manual es parte de la colección de manuales GNOME distribuidos bajo
la GFDL. Si quiere distribuir este manual separadamente de la colección, puede hacerlo añadiendo una copia
de la licencia al manual, tal como se describe en la sección 6 de la licencia.</p><p>Muchos de los nombres
usados por empresas para distinguir sus productos y servicios se mencionan como marcas comerciales. Donde
aparezcan dichos nombres en c
ualquier documentación GNOME, y para que los miembros del proyecto de documentación reconozcan dichas
marcas comerciales, dichos nombres se imprimen en mayúsculas o iniciales mayúsculas.</p><p>EL DOCUMENTO Y
LAS VERSIONES MODIFICADAS DEL MISMO SE PROPORCIONAN CON SUJECIÓN A LOS TÉRMINOS DE LA GFDL, QUEDANDO BIEN
ENTENDIDO, ADEMÁS, QUE: </p><div class="orderedlist"><ol class="orderedlist" type="1"><li
class="listitem"><p>EL DOCUMENTO SE ENTREGA "TAL CUAL", SIN GARANTÍA DE NINGÚN TIPO, NI EXPLÍCITA NI
IMPLÍCITA INCLUYENDO, SIN LIMITACIÓN, GARANTÍA DE QUE EL DOCUMENTO O VERSIÓN MODIFICADA DE ÉSTE CAREZCA
DE DEFECTOS EN EL MOMENTO DE SU VENTA, SEA ADECUADO A UN FIN CONCRETO O INCUMPLA ALGUNA NORMATIVA. TODO EL
RIESGO RELATIVO A LA CALIDAD, PRECISIÓN Y UTILIDAD DEL DOCUMENTO O SU VERSIÓN MODIFICADA RECAE EN USTED. SI
CUALQUIER DOCUMENTO O VERSIÓN MODIFICADA DE AQUÉL RESULTARA DEFECTUOSO EN CUALQUIER ASPECTO, USTED (Y NO EL
REDACTOR INICIAL, A
UTOR O AUTOR DE APORTACIONES) ASUMIRÁ LOS COSTES DE TODA REPARACIÓN, MANTENIMIENTO O CORRECCIÓN
NECESARIOS. ESTA EXENCIÓN DE RESPONSABILIDAD SOBRE LA GARANTÍA ES UNA PARTE ESENCIAL DE ESTA LICENCIA. NO
SE AUTORIZA EL USO DE NINGÚN DOCUMENTO NI VERSIÓN MODIFICADA DE ÉSTE POR EL PRESENTE, SALVO DENTRO DEL
CUMPLIMIENTO DE LA EXENCIÓN DE RESPONSABILIDAD;Y</p></li><li class="listitem"><p>BAJO NINGUNA CIRCUNSTANCIA
NI BAJO NINGUNA TEORÍA LEGAL, SEA POR ERROR (INCLUYENDO NEGLIGENCIA), CONTRATO O DE ALGÚN OTRO MODO, EL
AUTOR, EL ESCRITOR INICIAL, CUALQUIER CONTRIBUIDOR, O CUALQUIER DISTRIBUIDOR DEL DOCUMENTO O VERSIÓN
MODIFICADA DEL DOCUMENTO, O CUALQUIER PROVEEDOR DE CUALQUIERA DE ESAS PARTES, SERÁ RESPONSABLE ANTE NINGUNA
PERSONA POR NINGÚN DAÑO DIRECTO, INDIRECTO, ESPECIAL, INCIDENTAL O DERIVADO DE NINGÚN TIPO, INCLUYENDO,
SIN LIMITACIÓN DAÑOS POR PÉRDIDA DE MERCANCÍAS, PARO TÉCNICO, FALLO INFORMÁTICO O MAL FUNCIONAMIENTO O
CUALQUIER OT
RO POSIBLE DAÑO O PÉRDIDAS DERIVADAS O RELACIONADAS CON EL USO DEL DOCUMENTO O SUS VERSIONES MODIFICADAS,
AUNQUE DICHA PARTE HAYA SIDO INFORMADA DE LA POSIBILIDAD DE QUE SE PRODUJESEN DICHOS
DAÑOS.</p></li></ol></div></div></div><div><div class="legalnotice"><a name="idm45508419861488"></a><p
class="legalnotice-title"><b>Comentarios</b></p><p>Para informar de un fallo, o hacer alguna sugerencia sobre
la aplicación <span class="application">herramienta matemática Genius</span>, o este manual, siga las
instrucciones en la <a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">página web de
Genius</a> o envíe un correo electrónico a <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code>.</p></div></div><div><div class="revhistory"><table
style="border-style:solid; width:100%;" summary="Historial de revisiones"><tr><th align="left" valign="top"
colspan="2"><b>Historial de revisiones</b></th></tr><tr><td align="l
eft">Revisión 0.2</td><td align="left">Septiembre 2016</td></tr><tr><td align="left" colspan="2">
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Manual de
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Manual de la herramienta matemática Genius."><link rel="home" href="index.html" title="Manual de
Genius"><link rel="next" href="ch01.html" title="Capítulo 1. Introducción"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Manual de Genius</th></tr><tr><td width="20%"
align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Siguiente</a></td></tr></table><hr></div><div lang="es" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Manual de Genius</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><sp
an class="firstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Universidad del estado de Oklahoma<br></span><div class="address"><p> <code
class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code>
</p></div></div></div><div class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">Universidad de
Queensland, Australia<br></span><div class="address"><p> <code class="email"><<a class="email"
href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">Este manual describe la versión 1.0.22 de
Genius.</p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2011 Daniel
Mustieles (d
aniel mustieles gmail com)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Se concede
permiso para copiar, distribuir o modificar este documento según las condiciones de la GNU Free
Documentation License (GFDL), Versión 1.1 o cualquier versión posterior publicada por la Free Software
Foundation sin Secciones invariantes, Textos de portada y Textos de contraportada. Encontrará una copia de
la GFDL en este <a class="ulink" href="ghelp:fdl" target="_top">enlace</a> o en el archivo COPYING-DOCS
distribuido con este manual.</p><p>Este manual es parte de la colección de manuales GNOME distribuidos bajo
la GFDL. Si quiere distribuir este manual separadamente de la colección, puede hacerlo añadiendo una copia
de la licencia al manual, tal como se describe en la sección 6 de la licencia.</p><p>Muchos de los nombres
usados por empresas para distinguir sus productos y servicios se mencionan como marcas comerciales. Donde
aparezcan dichos nombres en c
ualquier documentación GNOME, y para que los miembros del proyecto de documentación reconozcan dichas
marcas comerciales, dichos nombres se imprimen en mayúsculas o iniciales mayúsculas.</p><p>EL DOCUMENTO Y
LAS VERSIONES MODIFICADAS DEL MISMO SE PROPORCIONAN CON SUJECIÓN A LOS TÉRMINOS DE LA GFDL, QUEDANDO BIEN
ENTENDIDO, ADEMÁS, QUE: </p><div class="orderedlist"><ol class="orderedlist" type="1"><li
class="listitem"><p>EL DOCUMENTO SE ENTREGA "TAL CUAL", SIN GARANTÍA DE NINGÚN TIPO, NI EXPLÍCITA NI
IMPLÍCITA INCLUYENDO, SIN LIMITACIÓN, GARANTÍA DE QUE EL DOCUMENTO O VERSIÓN MODIFICADA DE ÉSTE CAREZCA
DE DEFECTOS EN EL MOMENTO DE SU VENTA, SEA ADECUADO A UN FIN CONCRETO O INCUMPLA ALGUNA NORMATIVA. TODO EL
RIESGO RELATIVO A LA CALIDAD, PRECISIÓN Y UTILIDAD DEL DOCUMENTO O SU VERSIÓN MODIFICADA RECAE EN USTED. SI
CUALQUIER DOCUMENTO O VERSIÓN MODIFICADA DE AQUÉL RESULTARA DEFECTUOSO EN CUALQUIER ASPECTO, USTED (Y NO EL
REDACTOR INICIAL, A
UTOR O AUTOR DE APORTACIONES) ASUMIRÁ LOS COSTES DE TODA REPARACIÓN, MANTENIMIENTO O CORRECCIÓN
NECESARIOS. ESTA EXENCIÓN DE RESPONSABILIDAD SOBRE LA GARANTÍA ES UNA PARTE ESENCIAL DE ESTA LICENCIA. NO
SE AUTORIZA EL USO DE NINGÚN DOCUMENTO NI VERSIÓN MODIFICADA DE ÉSTE POR EL PRESENTE, SALVO DENTRO DEL
CUMPLIMIENTO DE LA EXENCIÓN DE RESPONSABILIDAD;Y</p></li><li class="listitem"><p>BAJO NINGUNA CIRCUNSTANCIA
NI BAJO NINGUNA TEORÍA LEGAL, SEA POR ERROR (INCLUYENDO NEGLIGENCIA), CONTRATO O DE ALGÚN OTRO MODO, EL
AUTOR, EL ESCRITOR INICIAL, CUALQUIER CONTRIBUIDOR, O CUALQUIER DISTRIBUIDOR DEL DOCUMENTO O VERSIÓN
MODIFICADA DEL DOCUMENTO, O CUALQUIER PROVEEDOR DE CUALQUIERA DE ESAS PARTES, SERÁ RESPONSABLE ANTE NINGUNA
PERSONA POR NINGÚN DAÑO DIRECTO, INDIRECTO, ESPECIAL, INCIDENTAL O DERIVADO DE NINGÚN TIPO, INCLUYENDO,
SIN LIMITACIÓN DAÑOS POR PÉRDIDA DE MERCANCÍAS, PARO TÉCNICO, FALLO INFORMÁTICO O MAL FUNCIONAMIENTO O
CUALQUIER OT
RO POSIBLE DAÑO O PÉRDIDAS DERIVADAS O RELACIONADAS CON EL USO DEL DOCUMENTO O SUS VERSIONES MODIFICADAS,
AUNQUE DICHA PARTE HAYA SIDO INFORMADA DE LA POSIBILIDAD DE QUE SE PRODUJESEN DICHOS
DAÑOS.</p></li></ol></div></div></div><div><div class="legalnotice"><a name="idm51"></a><p
class="legalnotice-title"><b>Comentarios</b></p><p>Para informar de un fallo, o hacer alguna sugerencia sobre
la aplicación <span class="application">herramienta matemática Genius</span>, o este manual, siga las
instrucciones en la <a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">página web de
Genius</a> o envíe un correo electrónico a <code class="email"><<a class="email" href="mailto:jirka 5z
com">jirka 5z com</a>></code>.</p></div></div><div><div class="revhistory"><table
style="border-style:solid; width:100%;" summary="Historial de revisiones"><tr><th align="left" valign="top"
colspan="2"><b>Historial de revisiones</b></th></tr><tr><td align="left">Revisi�
�n 0.2</td><td align="left">Septiembre 2016</td></tr><tr><td align="left" colspan="2">
<p class="author">Jiri (George) Lebl <code class="email"><<a class="email"
href="mailto:jirka 5z com">jirka 5z com</a>></code></p>
</td></tr></table></div></div><div><div class="abstract"><p
class="title"><b>Resumen</b></p><p>Manual de la herramienta matemática
Genius.</p></div></div></div><hr></div><div class="toc"><p><b>Tabla de contenidos</b></p><dl
class="toc"><dt><span class="chapter"><a href="ch01.html">1. Introducción</a></span></dt><dt><span
class="chapter"><a href="ch02.html">2. Primeros pasos</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch02.html#genius-to-start">Para iniciar la <span class="application">herramienta matemática
Genius</span></a></span></dt><dt><span class="sect1"><a href="ch02s02.html">Al iniciar
Genius</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch03.html">3. Uso
básico</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch03.html#genius-usage-workarea">Usar el área
de trabajo</a></span></dt><dt><span class="sect1"><a href="ch03s02.html">Crear un programa
nuevo</a></span></dt><dt><span class="sect1"><a href="ch03s03.html">Abrir y ejecuta
r un programa</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch04.html">4.
Dibujar</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch04.html#genius-line-plots">Trazado de
líneas</a></span></dt><dt><span class="sect1"><a href="ch04s02.html">Gráficos
paramétricos</a></span></dt><dt><span class="sect1"><a href="ch04s03.html">Dibujos de campos de
inclinación</a></span></dt><dt><span class="sect1"><a href="ch04s04.html">Gráficos de campos de
vectores</a></span></dt><dt><span class="sect1"><a href="ch04s05.html">Gráficos de
superficie</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch05.html">5. Conceptos de
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch05.html#genius-gel-values">Valores</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-numbers">Números</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-booleans">Booleanos</a></span></dt><dt><span class="sect2"><a hr
ef="ch05.html#genius-gel-values-strings">Cadenas</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-null">Nulo</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s02.html">Usar variables</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-setting">Configurar variables</a></span></dt><dt><span
class="sect2"><a href="ch05s02.html#genius-gel-variables-built-in">Variables
integradas</a></span></dt><dt><span class="sect2"><a href="ch05s02.html#genius-gel-previous-result">Resultado
de la variable anterior</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch05s03.html">Usar
funciones</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-defining">Definir funciones</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-variable-argument-lists">Listas de argumentos de
variables</a></span></dt><dt><span class="sect2"><a href="ch05s03.html#gen
ius-gel-functions-passing-functions">Pasar funciones a funciones</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-operations">Operaciones con
funciones</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s04.html">Separador</a></span></dt><dt><span class="sect1"><a
href="ch05s05.html">Comentarios</a></span></dt><dt><span class="sect1"><a href="ch05s06.html">Evaluación
modular</a></span></dt><dt><span class="sect1"><a href="ch05s07.html">Lista de operadores
GEL</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch06.html">6. Programar con
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch06.html#genius-gel-conditionals">Condicionales</a></span></dt><dt><span class="sect1"><a
href="ch06s02.html">Bucles</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-while">Bucles «while»</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-for">Bucles «for»</a></s
pan></dt><dt><span class="sect2"><a href="ch06s02.html#genius-gel-loops-foreach">Bucles
«foreach»</a></span></dt><dt><span class="sect2"><a href="ch06s02.html#genius-gel-loops-break-continue">Parar
y continuar</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch06s03.html">Sumas y
productos</a></span></dt><dt><span class="sect1"><a href="ch06s04.html">Operadores de
comparación</a></span></dt><dt><span class="sect1"><a href="ch06s05.html">Variables globales y ámbito de
variables</a></span></dt><dt><span class="sect1"><a href="ch06s06.html">Variables de
parámetros</a></span></dt><dt><span class="sect1"><a href="ch06s07.html">Retorno</a></span></dt><dt><span
class="sect1"><a href="ch06s08.html">Referencias</a></span></dt><dt><span class="sect1"><a
href="ch06s09.html">Lvalues</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch07.html">7.
Programación avanzada con GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch07.html#genius-gel-error-handl
ing">Control de errores</a></span></dt><dt><span class="sect1"><a href="ch07s02.html">Sintaxis de nivel
superior</a></span></dt><dt><span class="sect1"><a href="ch07s03.html">Devolver
funciones</a></span></dt><dt><span class="sect1"><a href="ch07s04.html">Variables locales
verdaderas</a></span></dt><dt><span class="sect1"><a href="ch07s05.html">Procedimiento de inicio de
GEL</a></span></dt><dt><span class="sect1"><a href="ch07s06.html">Cargar
programas</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch08.html">8. Matrices en
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch08.html#genius-gel-matrix-support">Introducir
matrices</a></span></dt><dt><span class="sect1"><a href="ch08s02.html">Conjugada de la traspuesta y operador
de trasposición</a></span></dt><dt><span class="sect1"><a href="ch08s03.html">Álgebra
lineal</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch09.html">9. Polinomios en
GEL</a></span></dt><dd><dl><dt><span class="s
ect1"><a href="ch09.html#genius-gel-polynomials-using">Usar polinomios</a></span></dt></dl></dd><dt><span
class="chapter"><a href="ch10.html">10. Teoría de conjuntos en GEL</a></span></dt><dd><dl><dt><span
class="sect1"><a href="ch10.html#genius-gel-sets-using">Usar conjuntos</a></span></dt></dl></dd><dt><span
class="chapter"><a href="ch11.html">11. Lista de funciones GEL</a></span></dt><dd><dl><dt><span
class="sect1"><a href="ch11.html#genius-gel-function-list-commands">Comandos</a></span></dt><dt><span
class="sect1"><a href="ch11s02.html">Básico</a></span></dt><dt><span class="sect1"><a
href="ch11s03.html">Parámetros</a></span></dt><dt><span class="sect1"><a
href="ch11s04.html">Constantes</a></span></dt><dt><span class="sect1"><a
href="ch11s05.html">Numérico</a></span></dt><dt><span class="sect1"><a
href="ch11s06.html">Trigonometría</a></span></dt><dt><span class="sect1"><a href="ch11s07.html">Teoría de
números</a></span></dt><dt><span class="sect1"><a href="c
h11s08.html">Manipulación de matrices</a></span></dt><dt><span class="sect1"><a
href="ch11s09.html">Álgebra lineal</a></span></dt><dt><span class="sect1"><a
href="ch11s10.html">Combinatoria</a></span></dt><dt><span class="sect1"><a
href="ch11s11.html">Cálculo</a></span></dt><dt><span class="sect1"><a
href="ch11s12.html">Funciones</a></span></dt><dt><span class="sect1"><a href="ch11s13.html">Resolución de
ecuaciones</a></span></dt><dt><span class="sect1"><a
href="ch11s14.html">Estadísticas</a></span></dt><dt><span class="sect1"><a
href="ch11s15.html">Polinomios</a></span></dt><dt><span class="sect1"><a href="ch11s16.html">Teoría de
conjuntos</a></span></dt><dt><span class="sect1"><a href="ch11s17.html">Álgebra
conmutativa</a></span></dt><dt><span class="sect1"><a href="ch11s18.html">Miscelánea</a></span></dt><dt><span
class="sect1"><a href="ch11s19.html">Operaciones simbólicas</a></span></dt><dt><span class="sect1"><a
href="ch11s20.html">Dibujar</a></span></dt
</dl></dd><dt><span class="chapter"><a href="ch12.html">12. Programas de ejemplo en
GEL</a></span></dt><dt><span class="chapter"><a href="ch13.html">13.
Configuración</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch13.html#genius-prefs-output">Salida</a></span></dt><dt><span class="sect1"><a
href="ch13s02.html">Precisión</a></span></dt><dt><span class="sect1"><a
href="ch13s03.html">Terminal</a></span></dt><dt><span class="sect1"><a
href="ch13s04.html">Memoria</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch14.html">14.
Acerca de la <span class="application">herramienta matemática Genius</span></a></span></dt></dl></div><div
class="list-of-figures"><p><b>Lista de figuras</b></p><dl><dt>2.1. <a
href="ch02s02.html#mainwindow-fig">Ventana de la <span class="application">herramienta matemática
Genius</span></a></dt><dt>4.1. <a href="ch04.html#lineplot-fig">Crear una ventana de
dibujo</a></dt><dt>4.2. <a href="ch04.html#lineplot2-fig">Ventana de dibu
jo</a></dt><dt>4.3. <a href="ch04s02.html#paramplot-fig">Pestaña dibujo paramétrico</a></dt><dt>4.4. <a
href="ch04s02.html#paramplot2-fig">Gráfico paramétrico</a></dt><dt>4.5. <a
href="ch04s05.html#surfaceplot-fig">Gráfico de superficie</a></dt></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left">
</td><td width="20%" align="center"> </td><td width="40%" align="right"> <a accesskey="n"
href="ch01.html">Siguiente</a></td></tr><tr><td width="40%" align="left" valign="top"> </td><td width="20%"
align="center"> </td><td width="40%" align="right" valign="top"> Capítulo 1.
Introducción</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch05s07.html b/help/fr/html/ch05s07.html
index d214a3b..ac27302 100644
--- a/help/fr/html/ch05s07.html
+++ b/help/fr/html/ch05s07.html
@@ -20,7 +20,15 @@ returns 3.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>
Element by element division. Same as <strong class="userinput"><code>a/b</code></strong>
for
numbers, but operates element by element on matrices.
- </p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Division arrière. C'est donc la même chose que
<strong class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Division arrière élément par
élément.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>L'opérateur modulo. Cela n'active pas le mode
d'<a class="link" href="ch05s06.html" title="Évaluation modulaire">évaluation modulaire</a>, mais renvoie
juste le reste de <strong class="userinput"><code>a/b</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>L'opérateur modulo élément par élément.
Renvoie le reste après la division entière élément par élément de <strong
class="userinput"><code>a./b</code></strong>.</p></dd><dt><span class="term
"><strong class="userinput"><code>a mod b</code></strong></span></dt><dd><p>Opérateur d'évaluation
modulaire. L'expression <code class="varname">a</code> est évaluée modulo <code class="varname">b</code>.
Consultez <a class="xref" href="ch05s06.html" title="Évaluation modulaire">la section intitulée « Évaluation
modulaire »</a>. Certaines fonctions et opérateurs se comportent différemment modulo un
entier.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Opérateur factoriel. Il s'agit de <strong
class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Opérateur double factoriel. Il s'agit de
<strong class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Division arrière. C'est donc la même chose que
<strong class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Division arrière élément par
élément.</p></dd><dt><span class="term"><strong class="userinput"><code>a%b</code></strong></span></dt><dd><p>
+ The mod operator. This does not turn on the <a class="link" href="ch05s06.html"
title="Évaluation modulaire">modular mode</a>, but
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
+ </p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>Opérateur d'évaluation modulaire. L'expression <code
class="varname">a</code> est évaluée modulo <code class="varname">b</code>. Consultez <a class="xref"
href="ch05s06.html" title="Évaluation modulaire">la section intitulée « Évaluation modulaire »</a>. Certaines
fonctions et opérateurs se comportent différemment modulo un entier.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Opérateur factoriel. Il s'agit de <strong
class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Opérateur double factoriel. Il s'agit de
<strong class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p>
Equality operator.
Returns <code class="constant">true</code> or <code class="constant">false</code>
depending on <code class="varname">a</code> and <code class="varname">b</code> being equal or
not.
@@ -37,21 +45,21 @@ returns 3.
greater than or equal to
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
- (can also be combine with the greater than operator).
+ (and they can also be combined with the greater than operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
Less than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
less than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
- (can also be combine with the less than or equal to operator).
+ (they can also be combined with the less than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
Greater than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
greater than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
- (can also be combine with the greater than or equal to operator).
+ (they can also be combined with the greater than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Opérateur comparaison. Si <code
class="varname">a</code> est égal à <code class="varname">b</code>, cela renvoie 0, si <code
class="varname">a</code> est inférieur à <code class="varname">b</code>, cela renvoie -1 et si <code
class="varname">a </code> est supérieur à <code class="varname">b</code>, cela renvoie 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a and b</code></strong></span></dt><dd><p>
Logical and. Returns true if both
<code class="varname">a</code> and <code class="varname">b</code> are true,
@@ -65,12 +73,12 @@ returns 3.
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
Logical xor.
- Returns true exactly one of
+ Returns true if exactly one of
<code class="varname">a</code> or <code class="varname">b</code> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
- Logical not. Returns the logical negation of <code class="varname">a</code>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>
@@ -85,7 +93,12 @@ returns 3.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>
Matrix transpose, does not conjugate the entries. That is,
the i,j element of <code class="varname">a</code> becomes the j,i element of <strong
class="userinput"><code>a.'</code></strong>.
- </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>Renvoie l'élément ligne <code
class="varname">b</code> et colonne <code class="varname">c</code> d'une matrice. Si <code
class="varname">b</code> et <code class="varname">c</code> sont des vecteurs alors cela renvoie les lignes et
les colonnes correspondantes, soit une sous-matrice.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Renvoie une ligne de matrice (ou plusieurs
lignes si <code class="varname">b</code> est un vecteur).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Comme ci-dessus.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Renvoie une colonne de
matrice (ou des colonnes si <code class="varname">c</code> est un vecteur).</p></dd><dt><span
class="term"><strong class="userinput"
<code>a@(:,c)</code></strong></span></dt><dd><p>Comme ci-dessus.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Renvoie un élément d'une matrice en le
traitant comme un vecteur. Cela parcourt la matrice dans le sens des lignes.</p></dd><dt><span
class="term"><strong class="userinput"><code>a:b</code></strong></span></dt><dd><p>Construit un vecteur
allant de <code class="varname">a</code> à <code class="varname">b</code> (ou indique une région ligne,
colonne pour l'opérateur <code class="literal">@</code>). Par exemple pour obtenir les lignes 2 à 4 de la
matrice <code class="varname">A</code>, nous pourrions faire </p><pre class="programlisting">A@(2:4,)
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>
+ Get element of a matrix in row <code class="varname">b</code> and column
+ <code class="varname">c</code>. If <code class="varname">b</code>,
+ <code class="varname">c</code> are vectors, then this gets the corresponding
+ rows, columns or submatrices.
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Renvoie une ligne de matrice (ou plusieurs
lignes si <code class="varname">b</code> est un vecteur).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Comme ci-dessus.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Renvoie une colonne de
matrice (ou des colonnes si <code class="varname">c</code> est un vecteur).</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Comme
ci-dessus.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Renvoie un élément d'une matrice en le
traitant comme un vecteur. Cela parcourt la matrice dans le sens des lignes.</p></dd><dt><span
class="term"><strong class="userinput"><code>a:b</code></strong></span></dt><dd><p>Con
struit un vecteur allant de <code class="varname">a</code> à <code class="varname">b</code> (ou indique une
région ligne, colonne pour l'opérateur <code class="literal">@</code>). Par exemple pour obtenir les lignes 2
à 4 de la matrice <code class="varname">A</code>, nous pourrions faire </p><pre
class="programlisting">A@(2:4,)
</pre><p> puisque <strong class="userinput"><code>2:4</code></strong> renvoie un vecteur <strong
class="userinput"><code>[2,3,4]</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b:c</code></strong></span></dt><dd><p>Construit un vecteur allant de <code
class="varname">a</code> à <code class="varname">c</code> avec un pas de <code class="varname">b</code>. Ce
qui donne par exemple </p><pre class="programlisting">genius> 1:2:9
=
`[1, 3, 5, 7, 9]
@@ -109,7 +122,12 @@ returns 3.
<strong class="userinput"><code>float(1:2/5:3)</code></strong> even gives you floating
point numbers and is ever so slightly more precise than
<strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
- </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Crée un nombre imaginaire (multiplie <code
class="varname">a</code> par le nombre imaginaire pur). Remarquez que normalement le nombre <code
class="varname">i</code> s'écrit <strong class="userinput"><code>1i</code></strong>. Donc le nombre ci-dessus
est égal à </p><pre class="programlisting">(a)*1i
- </pre></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Apostropher un identifiant afin qu'il ne soit
pas évalué. Ou apostropher une matrice afin qu'elle ne soit pas étendue.</p></dd><dt><span
class="term"><strong class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Échange la valeur
de <code class="varname">a</code> par la valeur de <code class="varname">b</code>. Pour le moment, ne
fonctionne pas sur des ensembles d'éléments de matrice. Renvoie <code class="constant">null</code>.
Disponible à partir de la version 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a</code></strong></span></dt><dd><p>Incrémente la variable <code
class="varname">a</code> de 1. Si <code class="varname">a</code> est une matrice alors incrémente chaque
élément. C'est équivalent à <strong class="userinput"><code>a=a+1</code></strong> mais est plus rapide.
Renvoie <code class="constant
">null</code>. Disponible à partir de la version 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a by b</code></strong></span></dt><dd><p>Incrémente la variable <code
class="varname">a</code> de <code class="varname">b</code>. Si <code class="varname">a</code> est une matrice
alors incrémente chaque élément. C'est équivalent à <strong class="userinput"><code>a=a+b</code></strong>
mais est plus rapide. Renvoie <code class="constant">null</code>. Disponible à partir de la version
1.0.13.</p></dd></dl></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Note</h3><p>L'opérateur @() rend l'opérateur : très utile. Grâce à lui, vous pouvez indiquer
des régions d'une matrice. Ainsi a@(2:4,6) sont les lignes 2,3,4 de la colonne 6 ou a@(,1:2) vous renvoie les
deux premières colonnes d'une matrice. Vous pouvez également attribuer un opérateur @() tant que la valeur de
droite est une matrice qui corr
espond en taille à la région ou si c'est n'importe quel autre type de valeur.</p></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Note</h3><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
+ written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
+ </p><pre class="programlisting">(a)*1i
+ </pre><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Apostropher un identifiant afin qu'il ne soit
pas évalué. Ou apostropher une matrice afin qu'elle ne soit pas étendue.</p></dd><dt><span
class="term"><strong class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Échange la valeur
de <code class="varname">a</code> par la valeur de <code class="varname">b</code>. Pour le moment, ne
fonctionne pas sur des ensembles d'éléments de matrice. Renvoie <code class="constant">null</code>.
Disponible à partir de la version 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a</code></strong></span></dt><dd><p>Incrémente la variable <code
class="varname">a</code> de 1. Si <code class="varname">a</code> est une matrice alors incrémente chaque
élément. C'est équivalent à <strong class="userinput"><code>a=a+1</code></strong> mais est plus rapide.
Renvoie <code class="const
ant">null</code>. Disponible à partir de la version 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a by b</code></strong></span></dt><dd><p>Incrémente la variable <code
class="varname">a</code> de <code class="varname">b</code>. Si <code class="varname">a</code> est une matrice
alors incrémente chaque élément. C'est équivalent à <strong class="userinput"><code>a=a+b</code></strong>
mais est plus rapide. Renvoie <code class="constant">null</code>. Disponible à partir de la version
1.0.13.</p></dd></dl></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Note</h3><p>L'opérateur @() rend l'opérateur : très utile. Grâce à lui, vous pouvez indiquer
des régions d'une matrice. Ainsi a@(2:4,6) sont les lignes 2,3,4 de la colonne 6 ou a@(,1:2) vous renvoie les
deux premières colonnes d'une matrice. Vous pouvez également attribuer un opérateur @() tant que la valeur de
droite est une matrice qui c
orrespond en taille à la région ou si c'est n'importe quel autre type de valeur.</p></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Note</h3><p>
The comparison operators (except for the <=> operator, which behaves normally), are not strictly
binary operators, they can in fact be grouped in the normal mathematical way, e.g.: (1<x<=y<5) is a
legal boolean expression and means just what it should, that is (1<x and x≤y and y<5)
</p></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Note</h3><p>L'opérateur unitaire « moins » agit de manière différente en fonction de l'endroit
où il apparaît. S'il apparaît devant un nombre, il est très prioritaire, s'il apparaît devant une expression,
il est moins prioritaire que les opérateurs puissance et factoriel. Par exemple, <strong
class="userinput"><code>-1^k</code></strong> est bien <strong class="userinput"><code>(-1)^k</code></strong>,
mais <strong class="userinput"><code>-foo(1)^k</code></strong> est bien <strong
class="userinput"><code>-(foo(1)^k)</code></strong>. En conséquence, faites attention à son utilisation et,
en cas de doute, ajoutez des parenthèses.</p></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch05s06.html">Précédent</a> </td><td width="20%" align="center"><a accesskey="u" href="ch05.htm
l">Niveau supérieur</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Suivant</a></td></tr><tr><td width="40%" align="left" valign="top">Évaluation modulaire
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Sommaire</a></td><td width="40%"
align="right" valign="top"> Chapitre 6. Programmation avec GEL</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch06s05.html b/help/fr/html/ch06s05.html
index 408d94c..616a766 100644
--- a/help/fr/html/ch06s05.html
+++ b/help/fr/html/ch06s05.html
@@ -1,4 +1,12 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Variables globales et
portée des variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manuel de Genius"><link rel="up" href="ch06.html" title="Chapitre 6. Programmation
avec GEL"><link rel="prev" href="ch06s04.html" title="Opérateurs de comparaison"><link rel="next"
href="ch06s06.html" title="Variables paramètres"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Variables globales et portée des variables</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Précédent</a> </td><th width="60%"
align="center">Chapitre 6. Programmation avec GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Suivant</a></td></tr></table><
hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Variables globales et portée des
variables</h2></div></div></div><p>GEL est un <a class="ulink"
href="http://fr.wikipedia.org/wiki/Port%C3%A9e_%28informatique%29"; target="_top">langage à portée
dynamique</a>. Nous allons expliquer ce que cela signifie ci-dessous. Les variables et les fonctions normales
sont à portée dynamique. Les exceptions sont les <a class="link" href="ch06s06.html" title="Variables
paramètres">variables paramètres</a> qui sont toujours globales.</p><p>Comme la plupart des langages de
programmation, GEL possède différents types de variables. Normalement lorsqu'une variable est définie dans
une fonction, elle est visible dans cette fonction et à partir de toutes les fonctions qui sont appelées
(tous les contextes supérieurs). Par exemple, supposons qu'une fonction <code class="function">f</code>
définit
une variable <code class="varname">a</code> puis appelle la fonction <code class="function">g</code>. Alors
la fonction <code class="function">g</code> peut faire référence à <code class="varname">a</code>. Mais dès
que la fonction <code class="function">f</code> est quittée, la variable <code class="varname">a</code>
disparaît de la portée. Par exemple, le code suivant affiche 5. La fonction <code class="function">g</code>
ne peut pas être appelée à partir du niveau supérieur (en dehors de <code class="function">f</code> puisque
<code class="varname">a</code> n'est pas défini). </p><pre class="programlisting">function f() = (a:=5; g());
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Variables globales et
portée des variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manuel de Genius"><link rel="up" href="ch06.html" title="Chapitre 6. Programmation
avec GEL"><link rel="prev" href="ch06s04.html" title="Opérateurs de comparaison"><link rel="next"
href="ch06s06.html" title="Variables paramètres"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Variables globales et portée des variables</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Précédent</a> </td><th width="60%"
align="center">Chapitre 6. Programmation avec GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Suivant</a></td></tr></table><
hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Variables globales et portée des variables</h2></div></div></div><p>
+ GEL is a
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ dynamically scoped language</a>. We will explain what this
+ means below. That is, normal variables and functions are dynamically
+ scoped. The exception are
+ <a class="link" href="ch06s06.html" title="Variables paramètres">parameter variables</a>,
+ which are always global.
+ </p><p>Comme la plupart des langages de programmation, GEL possède différents types de variables.
Normalement lorsqu'une variable est définie dans une fonction, elle est visible dans cette fonction et à
partir de toutes les fonctions qui sont appelées (tous les contextes supérieurs). Par exemple, supposons
qu'une fonction <code class="function">f</code> définit une variable <code class="varname">a</code> puis
appelle la fonction <code class="function">g</code>. Alors la fonction <code class="function">g</code> peut
faire référence à <code class="varname">a</code>. Mais dès que la fonction <code class="function">f</code>
est quittée, la variable <code class="varname">a</code> disparaît de la portée. Par exemple, le code suivant
affiche 5. La fonction <code class="function">g</code> ne peut pas être appelée à partir du niveau supérieur
(en dehors de <code class="function">f</code> puisque <code class="varname">a</code> n'est pas défini).
</p><pre class="pro
gramlisting">function f() = (a:=5; g());
function g() = print(a);
f();
</pre><p>Si vous définissez une variable à l'intérieur d'une fonction, elle va supplanter toutes variables
définies dans les fonctions appelantes. Par exemple, si nous modifions le code ci-dessus et écrivons :
</p><pre class="programlisting">function f() = (a:=5; g());
diff --git a/help/fr/html/ch07s02.html b/help/fr/html/ch07s02.html
index 421ce0c..18cca9a 100644
--- a/help/fr/html/ch07s02.html
+++ b/help/fr/html/ch07s02.html
@@ -3,10 +3,32 @@
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
- as just sequence of lines as if were entered on the command line.
+ as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
- parentheses).
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
</p><p>Le code suivant provoque une erreur lorsqu'il est saisi au niveau supérieur d'un programme
alors qu'il fonctionne très bien dans une fonction. </p><pre class="programlisting">if QuelqueChose() then
FaireQuelqueChose()
else
diff --git a/help/fr/html/ch11s04.html b/help/fr/html/ch11s04.html
index 86f52b8..4950c70 100644
--- a/help/fr/html/ch11s04.html
+++ b/help/fr/html/ch11s04.html
@@ -1,23 +1,26 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Constantes</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manuel de Genius"><link rel="up" href="ch11.html"
title="Chapitre 11. Liste des fonctions GEL"><link rel="prev" href="ch11s03.html" title="Paramètres"><link
rel="next" href="ch11s05.html" title="Nombres"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Constantes</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s03.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des
fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear
: both"><a name="genius-gel-function-list-constants"></a>Constantes</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Constante de Catalan, approximativement 0,915..., elle est définie
comme la série des termes <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong> où <code
class="varname">k</code> va de 0 à l'infini.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alias : <code class="function">gamma</code></p><p>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>Le
nombre d'or.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
- round and uniform.</p><p>Consultez <a class="ulink"
href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
+ round and uniform.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
is the exponential function
<a class="link" href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a>. It
is approximately
@@ -25,12 +28,12 @@
several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>Le nombre pi, c'est-à-dire le rapport de la circonférence d'un cercle sur son
diamètre. Il vaut approximativement 3.14159265359...</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Précédent</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Niveau supérieur</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch11s05.html">Suivant</a></td></tr><tr><td width="40%" align="left"
valign="top">Paramètres </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Sommaire</a></td><td width="40%" align="right" valign="top">
Nombres</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s05.html b/help/fr/html/ch11s05.html
index bdb77d4..cb56f39 100644
--- a/help/fr/html/ch11s05.html
+++ b/help/fr/html/ch11s05.html
@@ -5,17 +5,35 @@
to <strong class="userinput"><code>|x|</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
<a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
<a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld
(complex modulus)</a>
for more information.
- </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Remplace les très petits nombres par zéro.</p></dd><dt><span
class="term"><a name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alias : <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calcule le conjugué du nombre complexe <code class="varname">z</code>. Si
<code class="varname">z</code> est un vecteur ou une matrice, tous les éléments sont
conjugués.</p><p>Consultez <a class="ulink" href="http://fr.wikipedia.org/wiki/Conjugu%C3%A9";
target="_top">Wikipedia</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Renvoie le dénominateur d'un nombre rationnel.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/D%C3
%A9nominateurr" target="_top">Wikipedia</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Renvoie la partie fractionnelle d'un nombre.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/Partie_fractionnaire"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alias : <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>Consultez <a class="ulink" href="http://fr.wikipedia.org/wiki/Partie_imaginaire";
target="_top">Wikipedia</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuot
ient (m,n)</pre><p>Division sans reste.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(nbre)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
+ </p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Remplace les très petits nombres par zéro.</p></dd><dt><span
class="term"><a name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alias : <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calcule le conjugué du nombre complexe <code class="varname">z</code>. Si
<code class="varname">z</code> est un vecteur ou une matrice, tous les éléments sont conjugués.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Renvoie le dénominateur d'un nombre rationnel.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Renvoie la partie fractionnelle d'un nombre.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alias : <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Division sans reste.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(nbre)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
<strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
<strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (nbre)</pre><p>Vérifie si l'argument est potentiellement un nombre
rationnel complexe. C'est-à-dire si la partie réelle et la partie imaginaire sont fournies sous la forme de
nombres rationnels. Bien sûr, rationnel signifie simplement « non enregistré comme un nombre à virgule
flottante ».</p></dd><dt><span class="term"><a name="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre
class="synopsis">IsFloat (nbre)</pre><p>Check if argument is a real floating point number
(non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(nbre)</pre><p>Alias : <code class="function">IsComplexInteger</code></p><p>Check if argument is a possibly
complex int
eger. That is a complex integer is a number of
the form <strong class="userinput"><code>n+1i*m</code></strong> where <code
class="varname">n</code> and <code class="varname">m</code>
- are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(nbre)</pre><p>Vérifie si l'argument est un entier (non complexe).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (nbre)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (nbre)</pre><p>Alias : <code
class="function">IsNaturalNumber</code></p><p>Vérifie si l'argument est un entier réel positif. Notez que par
convention 0 n'est pas un nombre naturel.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(nbre)</pre><p>Vérifie si l'argument est un nombr
e rationnel (non complexe). Bien sûr, rationnel signifie simplement « non enregistré comme un nombre à
virgule flottante ».</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (nbre)</pre><p>Vérifie si
l'argument est un nombre réel.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator
(x)</pre><p>Renvoie le numérateur d'un nombre rationnel.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/Num%C3%A9rateur"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Alias : <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
3.</p><p>Consultez <a class="ulink" href="http://fr.wikipedia.org/wiki/Part
ie_r%C3%A9elle" target="_top">Wikipedia</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-Sign"></a>Sign</span></dt><dd><pre class="synopsis">Sign (x)</pre><p>Alias : <code
class="function">sign</code></p><p>Renvoie le signe d'un nombre. C'est-à-dire renvoie <code
class="literal">-1</code> si la valeur est négative, <code class="literal">0</code> si la valeur est nulle et
<code class="literal">1</code> si la valeur est positive. Si <code class="varname">x</code> est une grandeur
complexe alors <code class="function">Sign</code> renvoie la direction ou 0.</p></dd><dt><span
class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre class="synopsis">ceil (x)</pre><p>Alias
: <code class="function">Ceiling</code></p><p>Get the lowest integer more than or equal to <code
class="varname">n</code>. Examples:
+ are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(nbre)</pre><p>Vérifie si l'argument est un entier (non complexe).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (nbre)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (nbre)</pre><p>Alias : <code
class="function">IsNaturalNumber</code></p><p>Vérifie si l'argument est un entier réel positif. Notez que par
convention 0 n'est pas un nombre naturel.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(nbre)</pre><p>Vérifie si l'argument est un nombr
e rationnel (non complexe). Bien sûr, rationnel signifie simplement « non enregistré comme un nombre à
virgule flottante ».</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (nbre)</pre><p>Vérifie si
l'argument est un nombre réel.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator
(x)</pre><p>Renvoie le numérateur d'un nombre rationnel.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Alias : <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Alias : <code class="function">sign</code></p><p>Renvoie le signe d'un
nombre. C'est-à-dire renvoie <code class="literal">-1</code> si la valeur est négative, <code
class="literal">0</code> si la valeur est nulle et <code class="literal">1</code> si la valeur est positive.
Si <code class="varname">x</code> est une grandeur complexe alors <code class="function">Sign</code> renvoie
la direction ou 0.</p></dd><dt><span class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre
class="synopsis">ceil (x)</pre><p>Alias : <code class="function">Ceiling</code></p><p>Get the lowest integer
more than or equal to <code class="varname">n</code>. Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ceil(1.1)</code></strong>
= 2
<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1.1)</code></strong>
@@ -30,12 +48,12 @@ for more information.
exact arithmetic.
</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>La fonction exponentielle. C'est la fonction <strong
class="userinput"><code>e^x</code></strong> où <code class="varname">e</code> est la <a class="link"
href="ch11s04.html#gel-function-e">base du logarithme naturel</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Transforme le nombre en nombre à virgule flottante. C'est-à-dire la
représentation à virgule flottante du nombre <code class="varname">x</code>.</p></dd><dt><span
class="term"><a name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor
(x)</pre><p>Alias : <code class="function">Floor</code></p><p>Renvoie le plus grand entier inférieur ou égal
à <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-ln"></a>ln</span></dt><dd><pre class="synopsis">ln (x)</pre><p>Le logarithme naturel, le
logarithme de base <code class="varname">e</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logarithm of <code
class="varname">x</code> base <code class="varname">b</code> (calls <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> if in modulo
mode), if base is not given, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> is used.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logarithme base 10
de <code class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Alias : <code
class="function">lg</code></p><p>Logarithme base 2 de <code class="varname">x</code>.</p></dd><dt><span
class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="s
ynopsis">max (a,params...)</pre><p>Alias : <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Renvoie le maximum des arguments ou de la matrice.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,params...)</pre><p>Alias : <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Renvoie le minimum des arguments ou de la matrice.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(taille...)</pre><p>Génère aléatoirement des nombres flottants dans l'intervalle <code
class="literal">[0,1)</code>. Si taille est donnée alors une matrice (si deux nombres sont fournis) ou un
vecteur (si un seul est fourni) de la taille indiquée est renvoyé.</p></dd><dt><span class="term"><a
name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,taille...)</pre><p>Génère aléatoirement des
entiers dans l'intervalle <code class="literal">[0,max)</code>. Si taille est donné alors une matrice (si
deux nombres sont fournis) ou un vecteur (si un seul est fourni) de la taille indiquée est renvoyé. Par
exemple, </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
diff --git a/help/fr/html/ch11s06.html b/help/fr/html/ch11s06.html
index 5a9eb72..53d7137 100644
--- a/help/fr/html/ch11s06.html
+++ b/help/fr/html/ch11s06.html
@@ -1,13 +1,21 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonométrie</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manuel de Genius"><link rel="up" href="ch11.html"
title="Chapitre 11. Liste des fonctions GEL"><link rel="prev" href="ch11s05.html" title="Nombres"><link
rel="next" href="ch11s07.html" title="Théorie des nombres"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonométrie</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des
fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="ti
tle" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonométrie</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alias : <code
class="function">arccos</code></p><p>Fonction arccos (arc cosinus).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Alias : <code
class="function">arccosh</code></p><p>Fonction arccosh (cosinus hyperbolique inverse).</p></dd><dt><span
class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Alias
: <code class="function">arccot</code></p><p>Fonction arccot (cotangente inverse).</p></dd><dt><span
class="term"><a name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth
(x)</pre><p>Alias : <code class="function">arccoth</code></p><p>Fonction a
rccoth (cotangente hyperbolique inverse).</p></dd><dt><span class="term"><a
name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc (x)</pre><p>Alias : <code
class="function">arccsc</code></p><p>Inverse de la fonction cosécante.</p></dd><dt><span class="term"><a
name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch (x)</pre><p>Alias : <code
class="function">arccsch</code></p><p>Inverse de la fonction cosécante hyperbolique.</p></dd><dt><span
class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Alias
: <code class="function">arcsec</code></p><p>Inverse de la fonction sécante.</p></dd><dt><span
class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech
(x)</pre><p>Alias : <code class="function">arcsech</code></p><p>Inverse de la fontion sécante
hyperbolique.</p></dd><dt><span class="term"><a name="gel-function-asin"></a>asin</span></dt><dd
<pre class="synopsis">asin (x)</pre><p>Alias : <code class="function">arcsin</code></p><p>La fonction
arcsin (sinus inverse).</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Alias : <code
class="function">arcsinh</code></p><p>Fonction arcsinh (sinus hyperbolique inverse).</p></dd><dt><span
class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan
(x)</pre><p>Alias : <code class="function">arctan</code></p><p>Calcule la fonction arctangente (tangente
inverse).</p><p>Consultez <a class="ulink" href="http://fr.wikipedia.org/wiki/Arctangente";
target="_top">Wikipedia</a> or <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-atanh"></a>atanh</span></dt><dd><pre class="synopsis">atanh (x)</pre><p>Alias : <code
class="function">arctanh</code>
</p><p>Fonction arctanh (tangente hyperbolique inverse).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alias : <code
class="function">arctan2</code></p><p>Calculates the arctan2 function. If
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonométrie</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manuel de Genius"><link rel="up" href="ch11.html"
title="Chapitre 11. Liste des fonctions GEL"><link rel="prev" href="ch11s05.html" title="Nombres"><link
rel="next" href="ch11s07.html" title="Théorie des nombres"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonométrie</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des
fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="ti
tle" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonométrie</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alias : <code
class="function">arccos</code></p><p>Fonction arccos (arc cosinus).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Alias : <code
class="function">arccosh</code></p><p>Fonction arccosh (cosinus hyperbolique inverse).</p></dd><dt><span
class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Alias
: <code class="function">arccot</code></p><p>Fonction arccot (cotangente inverse).</p></dd><dt><span
class="term"><a name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth
(x)</pre><p>Alias : <code class="function">arccoth</code></p><p>Fonction a
rccoth (cotangente hyperbolique inverse).</p></dd><dt><span class="term"><a
name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc (x)</pre><p>Alias : <code
class="function">arccsc</code></p><p>Inverse de la fonction cosécante.</p></dd><dt><span class="term"><a
name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch (x)</pre><p>Alias : <code
class="function">arccsch</code></p><p>Inverse de la fonction cosécante hyperbolique.</p></dd><dt><span
class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Alias
: <code class="function">arcsec</code></p><p>Inverse de la fonction sécante.</p></dd><dt><span
class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech
(x)</pre><p>Alias : <code class="function">arcsech</code></p><p>Inverse de la fontion sécante
hyperbolique.</p></dd><dt><span class="term"><a name="gel-function-asin"></a>asin</span></dt><dd
<pre class="synopsis">asin (x)</pre><p>Alias : <code class="function">arcsin</code></p><p>La fonction
arcsin (sinus inverse).</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Alias : <code
class="function">arcsinh</code></p><p>Fonction arcsinh (sinus hyperbolique inverse).</p></dd><dt><span
class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan
(x)</pre><p>Alias : <code class="function">arctan</code></p><p>Calcule la fonction arctangente (tangente
inverse).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alias : <code class="function">arctanh</code></p><p>Fonction arctanh
(tangente hyperbolique inverse).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alias : <code
class="function">arctan2</code></p><p>Calculates the arctan2 function. If
<strong class="userinput"><code>x>0</code></strong> then it returns
<strong class="userinput"><code>atan(y/x)</code></strong>. If <strong
class="userinput"><code>x<0</code></strong>
then it returns <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>.
When <strong class="userinput"><code>x=0</code></strong> it returns <strong
class="userinput"><code>sign(y) *
pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> returns 0
rather than failing.
- </p><p>Consultez <a class="ulink" href="http://fr.wikipedia.org/wiki/Atan2";
target="_top">Wikipedia</a> ou <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-cos"></a>cos</span></dt><dd><pre class="synopsis">cos (x)</pre><p>Calcule la fonction
cosinus.</p><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Calcule la fonction cosinus.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Calcule la fonction cosinus hyperbolique.</p><p>
See
@@ -15,7 +23,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>Fonction cotangente.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Fonction cotangente hyperbolique.</p><p>
See
@@ -23,7 +31,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>Fonction cosécante.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Fonction cosécante hyperbolique.</p><p>
See
@@ -31,7 +39,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Fonction sécante.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Fonction sécante hyperbolique.</p><p>
See
@@ -39,7 +47,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Calcule la fonction sinus.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Calcule la fonction sinus hyperbolique.</p><p>
See
@@ -47,7 +55,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calcule la fonction tangente.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Fonction tangente hyperbolique.</p><p>
See
diff --git a/help/fr/html/ch11s07.html b/help/fr/html/ch11s07.html
index 0fffbd5..34e507f 100644
--- a/help/fr/html/ch11s07.html
+++ b/help/fr/html/ch11s07.html
@@ -3,25 +3,32 @@
<a class="ulink" href="https://en.wikipedia.org/wiki/Coprime_integers";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Renvoie le <code class="varname">n</code>-ième nombre de Bernoulli.</p><p>Consultez <a
class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a> ou <a
class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html"; target="_top">Mathworld</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alias : <code class="function">CRT</code></p><p>Recherche
<code class="varname">x</code> qui résout le système défini par le vecteur <code class="varname">a</code> et
modulo les éléments de <code class="varname">m</code>, en utilisant le théorème des restes chinois.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Renvoie le <code class="varname">n</code>-ième nombre de Bernoulli.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alias : <code class="function">CRT</code></p><p>Recherche
<code class="varname">x</code> qui résout le système défini par le vecteur <code class="varname">a</code> et
modulo les éléments de <code class="varname">m</code>, en utilisant le théorème des restes chinois.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Étant donné deux factorisations, donne la factorisation
du produit.</p><p>Consultez <a class="link"
href="ch11s07.html#gel-function-Factorize">Factorize</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Convertit un vecteur de valeurs indiquant les puissances de b en un nombre.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Convertit un nombre en un vecteur contenant les puissances des
éléments dans la base <code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synopsis">DiscreteLog
(n,b,q)</pre><p>
Calcule le logarithme discret de <code class="varname">n</code> base <code class="varname">b</code> dans
F<sub>q</sub>, le corps fini d'ordre <code class="varname">q</code> où <code class="varname">q</code> est un
nombre premier, en utilisant l'algorithme de Silver-Pohlig-Hellman.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Vérifie la divisibilité (si <code class="varname">m</code> divise
<code class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>Calcule
la fonction d'Euler phi, c'est-à-dire le nombre d'entiers compris entre 1 et <code class="varname">n</code>
qui sont premiers avec <code class="varname">n</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Renvoie <strong class="userinput"><code>n/d</code></strong> mais seulement si <code
class="varname">d</code> divise <code class="varname">n</code>. Si <code class="varname">d</code> ne divise
pas <code class="varname">n</code> alors cette fonction ne renvoie rien d'utile. Cette fonction est beaucoup
plus rapide pour les très grands nombres que l'opération <strong class="userinput"><code>n/d</code></strong>,
mais bien sûr utile seulement si vous savez que la division est exacte.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize
(n)</pre><p>Renvoie la factorisation d'un nombre sous la forme d'une matrice. La première ligne contient les
nombres premiers dans la factorisation (y compris 1) et la seconde ligne sont les puissances. Par exemple :
</p>
<pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>Factorize(11*11*13)</code></strong>
=
[1 11 13
- 1 2 1]</pre><p>Consultez <a class="ulink" href="http://fr.wikipedia.org/wiki/Factorisation";
target="_top">Wikipedia</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-Factors"></a>Factors</span></dt><dd><pre class="synopsis">Factors (n)</pre><p>Renvoie tous
les facteurs de <code class="varname">n</code> dans un vecteur. Cela inclut tous les facteurs non premiers
également mais aussi 1 et le nombre lui-même. Ainsi par exemple pour afficher tous les nombres parfaits (ceux
qui sont la somme de leurs facteurs) jusqu'au nombre 1000, vous pouvez écrire (ce n'est bien sûr pas
efficace) : </p><pre class="programlisting">for n=1 to 1000 do (
+ 1 2 1]</pre><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>Renvoie tous les facteurs de <code class="varname">n</code> dans un
vecteur. Cela inclut tous les facteurs non premiers également mais aussi 1 et le nombre lui-même. Ainsi par
exemple pour afficher tous les nombres parfaits (ceux qui sont la somme de leurs facteurs) jusqu'au nombre
1000, vous pouvez écrire (ce n'est bien sûr pas efficace) : </p><pre class="programlisting">for n=1 to 1000
do (
if MatrixSum (Factors(n)) == 2*n then
print(n)
)
@@ -31,7 +38,10 @@
and <code class="varname">s</code> as a vector if possible, <code class="constant">null</code>
otherwise.
<code class="varname">tries</code> specifies the number of tries before
giving up.
- </p><p>C'est une assez bonne factorisation si votre nombre est le produit de deux facteurs très
proches l'un de l'autre.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/M%C3%A9thode_de_factorisation_de_Fermat"; target="_top">Wikipedia</a> pour
plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Cherche le premier élément primitif dans F<sub>q</sub>,
le groupe fini d'ordre <code class="varname">q</code>. Bien sûr, <code class="varname">q</code> doit être
premier.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Cherche un élément primitif au hasard dans
F<sub>q</sub>, le groupe fini d'ordre <code class="varname">q</code> (q doit être premier).</p></dd><dt>
<span class="term"><a name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre
class="synopsis">IndexCalculus (n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of
n in F<sub>q</sub>, the finite
+ </p><p>C'est une assez bonne factorisation si votre nombre est le produit de deux facteurs très
proches l'un de l'autre.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Cherche le premier élément primitif dans F<sub>q</sub>,
le groupe fini d'ordre <code class="varname">q</code>. Bien sûr, <code class="varname">q</code> doit être
premier.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Cherche un élément primitif au hasard dans
F<sub>q</sub>, le groupe fini d'ordre <code class="varname">q</code> (q doit être premier).</p></dd><dt><span
class="term"><a name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre
class="synopsis">IndexCalculus (n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of
n in F<sub>q</sub>, the finite
group of order <code class="varname">q</code> (<code class="varname">q</code> a prime), using the
factor base <code class="varname">S</code>. <code class="varname">S</code> should be a column of
primes possibly with second column precalculated by
@@ -52,12 +62,16 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Vérifie si un nombre rationnel <code class="varname">m</code> est une puissance <code
class="varname">n</code>-ième parfaite. Consultez aussi <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> et <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Teste si un entier
est impair.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>Vé
rifie qu'un entier est un carré parfait d'un entier. Le nombre doit être un vrai entier. Les entiers
négatifs ne peuvent bien sûr jamais être des carrés de vrais entiers.</p></dd><dt><span class="term"><a
name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre class="synopsis">IsPrime (n)</pre><p>Teste la
primalité des entiers ; pour les nombres inférieurs à 2,5e10 la réponse est déterministe (si l'hypothèse de
Riemann est vérifiée). Pour des nombres plus grands, la probabilité d'une erreur de détermination dépend du
paramètre <a class="link" href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. C'est-à-dire la probabilité d'une erreur de détermination
vaut 1/4 à la puissance <code class="function">IsPrimeMillerRabinReps</code>. La valeur par défaut de 22 mène
à une probabilité d'environ 5.7e-14.</p><p>Si <code class="constant">false</code> (faux) est renvoyé, vous
êtes sûr que le n
ombre est composé. Si vous voulez être absolument certain d'avoir un nombre premier vous pouvez utiliser la
fonction <a class="link" href="ch11s07.html#gel-function-MillerRabinTestSure"><code
class="function">MillerRabinTestSure</code></a> mais cela peut prendre beaucoup plus de temps.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Vérifie si un nombre rationnel <code class="varname">m</code> est une puissance <code
class="varname">n</code>-ième parfaite. Consultez aussi <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> et <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Teste si un entier
est impair.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
+ Check an integer for being a perfect square of an integer. The number must
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
+ </p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>Teste la primalité des entiers ; pour les nombres inférieurs à 2,5e10 la
réponse est déterministe (si l'hypothèse de Riemann est vérifiée). Pour des nombres plus grands, la
probabilité d'une erreur de détermination dépend du paramètre <a class="link"
href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. C'est-à-dire la probabilité d'une erreur de détermination
vaut 1/4 à la puissance <code class="function">IsPrimeMillerRabinReps</code>. La valeur par défaut de 22 mène
à une probabilité d'environ 5.7e-14.</p><p>Si <code class="constant">false</code> (faux) est renvoyé, vous
êtes sûr que le nombre est composé. Si vous voulez être absolument certain d'avoir un nombre premier vous
pouvez utiliser la fonction <a class="link" href="ch11s07.html#gel-function-Miller
RabinTestSure"><code class="function">MillerRabinTestSure</code></a> mais cela peut prendre beaucoup plus de
temps.</p><p>
See
<a class="ulink" href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a>
for more information.
@@ -68,24 +82,24 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Teste si 2<sup>p</sup>-1 est un nombre premier de Mersenne en utilisant le test de Lucas-Lehmer.
Consultez aussi <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> et <a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Renvoie le <code class="varname">n</code>-ième nombre de Lucas.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Renvoie les puissances premières d'un
nombre.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>Renvoie un vecteur de nombres premiers de Mersenne qui est
une liste d'entiers positifs <code class="varname">p</code> tels que 2<sup>p</sup>-1 est entier. Consultez
aussi <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a> et
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre class="synopsis">MillerRabinTest
(n,reps)</pre><p>Utilise le test de primalité de Miller-Rabin sur <code class="varname">n</code>, en faisant
<code class="varname">reps</code> essais. La probabilité d'une erreur de détermination est <strong
class="userinput"><code>(1/4)^reps</code></strong>. Il est probablement préférable d'utiliser la fonction <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a> puisqu'elle
est plus rapide et meilleure pour les entiers les plus petits.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
@@ -94,7 +108,7 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
result is deterministic.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Renvoie l'inverse de n mod m.</p><p>Consultez <a class="ulink"
href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu
(n)</pre><p>Renvoie la fonction mu de Moebius évaluée dans <code class="varname">n</code>. C'est-à-dire
renvoie 0 si <code class="varname">n</code> n'est pas un produit de nombres premiers différents et <strong
class="userinput"><code>(-1)^k</code></strong> si c'est un produit de <code class="varname">k</code> nombres
premiers différents.</p><p>
diff --git a/help/fr/html/ch11s08.html b/help/fr/html/ch11s08.html
index b95ce6c..cd9b126 100644
--- a/help/fr/html/ch11s08.html
+++ b/help/fr/html/ch11s08.html
@@ -1,9 +1,12 @@
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matrices</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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header"><tr><th colspan="3" align="center">Manipulation de matrices</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s07.html">Précédent</a> </td><th width="60%" align="center">Chapitre
11. Liste des fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s09.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div class="tit
lepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Manipulation de matrices</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,fonc)</pre><p>Applique une fonction sur tous les éléments d'une matrice et renvoie une matrice de
résultats.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,fonc)</pre><p>Applique une fonction sur tous les éléments de 2
matrices (ou 1 valeur et 1 matrice) et renvoie une matrice de résultats.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf
(M)</pre><p>Extrait les colonnes de la matrice comme un vecteur horizontal.</p></dd><dt><span class="term"><a
name="gel-func
tion-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre class="synopsis">ComplementSubmatrix
(m,r,c)</pre><p>Supprime certaines lignes et colonnes d'une matrice.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Calcule la k-ième matrice composée de A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
- Count the number of zero columns in a matrix. For example
- once your column reduce a matrix you can use this to find
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
- </p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Supprime une colonne d'une matrice.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Supprime une ligne d'une matrice.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Extrait la diagonale de la matrice comme un vecteur colonne.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/Diagonale"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
+ </p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Supprime une colonne d'une matrice.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Supprime une ligne d'une matrice.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Extrait la diagonale de la matrice comme un vecteur colonne.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Dot_product"; target="_top">Wikipedia</a> or
@@ -18,14 +21,20 @@
<a class="ulink" href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vec,mtaille)</pre><p>Renvoie le complémentaire d'un vecteur d'indices. Le premier indice est toujours 1. Par
exemple pour le vecteur <strong class="userinput"><code>[2,3]</code></strong> et la taille <strong
class="userinput"><code>5</code></strong>, cela renvoie <strong
class="userinput"><code>[1,4,5]</code></strong>. Si <code class="varname">mtaille</code> vaut 0, cela renvoie
toujours <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal
(M)</pre><p>Indique si la matrice est diagonale.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Vérifie qu'une matrice est la matrice identité. Renvoie automatiquement <code
class="constant">false</code> (faux) si la matrice n'est pas carrée. Fonctionne également avec les nombres et
dans ce cas, c'est équivalent à <strong class="userinput"><code>x==1</code></strong>. Lorsque <code
class="varname">x</code> est <code class="constant">null</code>, il est considéré comme une matrice 0 par 0,
aucune erreur n'est générée et <code class="constant">false</code> (faux) est renvoyé.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Indique si une matrice est triangulaire inférieure,
c'est-à-dire que toutes les valeurs au dessus de la diagonale sont nulles.</p></dd><dt><span class="term"><a
name="gel-function-IsMatr
ixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger (M)</pre><p>Check if a
matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Vérifie si une matrice est non négative, c'est-à-dire que
chaque élément n'est pas négatif. Ne pas confondre les matrices positives avec les matrices définies
positives.</p><p>Consultez <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix";
target="_top">Wikipedia</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Vérifie si une matrice est positive, c'est-à-dire que chaque
élément est positif (et par conséquent réel), et en particulier qu'aucun élément n'est nul. Ne pas confondre
les matrices posi
tives avec les matrices définies positives</p><p>Consultez <a class="ulink"
href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Vérifie si une matrice est constituée de nombres rationnels
(non complexes).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Vérifie si une matrice est constituée de nombres réels (non complexes).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Vérifie si une matrice est carrée, c'est-à-dire que sa largeur
est égale à sa hauteur.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt
<dd><pre class="synopsis">IsUpperTriangular (M)</pre><p>Is a matrix upper triangular? That is, a matrix is
upper triangular if all the entries below the diagonal are zero.</p></dd><dt><span class="term"><a
name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Vérifie si une matrice est une matrice de nombres seulement. Beaucoup de fonctions internes
contrôlent cela. Les valeurs peuvent être n'importe quels nombres y compris des
complexes.</p></dd><dt><span class="term"><a name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre
class="synopsis">IsVector (v)</pre><p>Si l'argument est un vecteur horizontal ou vertical. Genius ne fait
pas de distinction entre une matrice et un vecteur, un vecteur est juste une matrice 1 par <code
class="varname">n</code> ou <code class="varname">n</code> par 1.</p></dd><dt><span class="term"><a
name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero (x)</
pre><p>Vérifie si une matrice est composée uniquement de zéros. Fonctionne également avec les nombres et
dans ce cas, c'est équivalent à <strong class="userinput"><code>x==0</code></strong>. Lorsque <code
class="varname">x</code> est <code class="constant">null</code>, il est considéré comme une matrice 0 par 0,
aucune erreur n'est générée et <code class="constant">true</code> (vrai) est renvoyé car la condition est
vide.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Renvoie une copie de la matrice <code class="varname">M</code> avec tous les éléments au dessus
de la diagonale mis à zéro.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,param...)</pre><p>Alias : <code class="function">diag</code></p><p>Make diagonal matrix from a vector.
Alternatively you can pass
+ </p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Vérifie qu'une matrice est la matrice identité. Renvoie automatiquement <code
class="constant">false</code> (faux) si la matrice n'est pas carrée. Fonctionne également avec les nombres et
dans ce cas, c'est équivalent à <strong class="userinput"><code>x==1</code></strong>. Lorsque <code
class="varname">x</code> est <code class="constant">null</code>, il est considéré comme une matrice 0 par 0,
aucune erreur n'est générée et <code class="constant">false</code> (faux) est renvoyé.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Indique si une matrice est triangulaire inférieure,
c'est-à-dire que toutes les valeurs au dessus de la diagonale sont nulles.</p></dd><dt><span class="term"><a
name="gel-function-IsMatr
ixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger (M)</pre><p>Check if a
matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Vérifie si une matrice est non négative, c'est-à-dire que
chaque élément n'est pas négatif. Ne pas confondre les matrices positives avec les matrices définies
positives.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Vérifie si une matrice est positive, c'est-à-dire que chaque
élément est positif (et par conséquent réel), et en particulier qu'aucun élément n'est nul. Ne pas confondre
les matrices positives avec les matrices définies positives</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Vérifie si une matrice est constituée de nombres rationnels
(non complexes).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Vérifie si une matrice est constituée de nombres réels (non complexes).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre
class="synopsis">IsMatrixSquare (M)</pre><p>Vérifie si une matrice est carrée, c'est-à-dire que sa largeur
est égale à sa hauteur.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Is a matrix upper triangular? That is, a matrix is upper
triangular if all the entries below the diagonal are zero.</p></
dd><dt><span class="term"><a name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre
class="synopsis">IsValueOnly (M)</pre><p>Vérifie si une matrice est une matrice de nombres seulement.
Beaucoup de fonctions internes contrôlent cela. Les valeurs peuvent être n'importe quels nombres y compris
des complexes.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Si
l'argument est un vecteur horizontal ou vertical. Genius ne fait pas de distinction entre une matrice et un
vecteur, un vecteur est juste une matrice 1 par <code class="varname">n</code> ou <code
class="varname">n</code> par 1.</p></dd><dt><span class="term"><a
name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero (x)</pre><p>Vérifie si une
matrice est composée uniquement de zéros. Fonctionne également avec les nombres et dans ce cas, c'est
équivalent à <strong class="userinput"><code>x==0
</code></strong>. Lorsque <code class="varname">x</code> est <code class="constant">null</code>, il est
considéré comme une matrice 0 par 0, aucune erreur n'est générée et <code class="constant">true</code> (vrai)
est renvoyé car la condition est vide.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Renvoie une copie de la matrice <code class="varname">M</code> avec tous les éléments au dessus
de la diagonale mis à zéro.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,param...)</pre><p>Alias : <code class="function">diag</code></p><p>Make diagonal matrix from a vector.
Alternatively you can pass
in the values to put on the diagonal as arguments. So
<strong class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> is the same as
<strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Construit un vecteur colonne à partir d'une matrice en mettant les colonnes les unes au dessus
des autres. Renvoie <code class="constant">null</code> si <code class="constant">null</code> est
fourni.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>Calcule et renvoie le produit de tous les éléments d'une matrice ou d'un
vecteur.</p></dd><dt><span class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre
class="synopsis">MatrixSum (A)</pre><p>Calcule et renvoie la somme de tous les éléments d'une matrice ou d'un
vecteur.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Calcule la somme du carré
de tous les éléments d'une matrice ou d'un vecteur.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Returns a row vector of the indices of nonzero columns in the matrix <code
class="varname">M</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Returns a row vector of the indices of nonzero elements in the vector <code
class="varname">v</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Retourne le produit tensoriel de deux vecteurs, c'est-à-dire que si on suppose que <code
class="varname">u</code> et <code class="varname">v</code> sont des vecteurs colonnes, alors le produit
tensoriel est <strong c
lass="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Inverse l'ordre des éléments d'un vecteur. Renvoie <code class="constant">null</code> si <code
class="constant">null</code> est fourni</p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Calcule la
somme pour chaque ligne d'une matrice et renvoie un vecteur colonne contenant le résultat.</p></dd><dt><span
class="term"><a name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre
class="synopsis">RowSumSquares (m)</pre><p>Calculate sum of squares of each row in a matrix and return a
vertical vector with the results.</p></dd><dt><span class="term"><a
name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre class="synopsis">RowsOf (M)</pre><p>Gets the rows
of a matrix as a vertical vector. Ea
ch element
of the vector is a horizontal vector that is the corresponding row of
diff --git a/help/fr/html/ch11s09.html b/help/fr/html/ch11s09.html
index 53689fa..1b40238 100644
--- a/help/fr/html/ch11s09.html
+++ b/help/fr/html/ch11s09.html
@@ -31,12 +31,12 @@ superdiagonal being all ones. It is the Jordan block matrix of one zero eigenva
<a class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Alias : <code class="function">eig</code></p><p>Renvoie les valeurs propres d'une matrice carrée.
Ne fonctionne actuellement que pour les matrices de taille inférieure ou égale à 4 par 4 ou pour les matrices
triangulaires (pour lesquelles les valeurs propres sont sur la diagonale).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &valeurspropres)</pre><pre class="synopsis">Eigenvectors
(M, &valeurpropres, &multiplicités)</pre><p>Renvoie les vecteurs propres d'une matrice carrée. Il est
possible en option d'obtenir les valeurs propres ainsi que leur multiplicité algébrique. Ne fonctionne
actuellement que pour les matrices 2x2.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Applique le procédé de Gram-Schmidt (aux colonnes) par rapport au produit scalaire donné par
<code class="varname">B</code>. Si <code class="varname">B</code> n'est pas fourni alors le produit hermitien
standard est utilisé. <code class="varname">B</code> peut être soit une forme sesquilinéaire à deux arguments
soit une matrice fournissant une forme sesquilinéaire. Les vecteurs seront orthogonaux par rapport à <code
class="varname">B</code>.</p><p>
@@ -92,7 +92,7 @@ superdiagonal being all ones. It is the Jordan block matrix of one zero eigenva
of two matrices.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
@@ -117,7 +117,7 @@ superdiagonal being all ones. It is the Jordan block matrix of one zero eigenva
the 1's diagonal on the lower matrix.
</p><p>Toutes les matrices ne possèdent pas de décomposition LU, par exemple <strong
class="userinput"><code>[0,1;1,0]</code></strong> n'en a pas. Dans ce cas, cette fonction renvoie <code
class="constant">false</code> (faux) et initialise <code class="varname">L</code> et <code
class="varname">U</code> à <code class="constant">null</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Renvoie le mineur <code class="varname">i</code>-<code
class="varname">j</code> d'une matrice.</p><p>
@@ -132,7 +132,7 @@ superdiagonal being all ones. It is the Jordan block matrix of one zero eigenva
</p></dd><dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre
class="synopsis">OrthogonalComplement (M)</pre><p>Renvoie le complément orthogonal de l'espace des
colonnes.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Return pivot columns of a matrix, that is columns that have a leading 1 in row reduced form.
Also returns the row where they occur.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Projection du vecteur <code class="varname">v</code> sur le sous-espace <code
class="varname">W</code> par rapport au produit scalaire donné par <code class="varname">B</code>. Si <code
class="varname">B</code> n'est pas fourni alors le produit hermitien standard est utilisé. <code
class="varname">B</code> peut être s
oit une forme sesquilinéaire à deux arguments soit une matrice fournissant une forme
sesquilinéaire.</p></dd><dt><span class="term"><a
name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre class="synopsis">QRDecomposition
(A, Q)</pre><p>Calcule la décomposition QR d'une matrice carrée <code class="varname">A</code>, renvoie la
matrice triangulaire supérieure <code class="varname">R</code> et définit <code class="varname">Q</code>
comme la matrice orthogonale (unitaire). <code class="varname">Q</code> doit être une référence. Si vous ne
voulez pas qu'elle soit renvoyée, utilisez <code class="constant">null</code>. Par exemple : </p><pre
class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>R =
QRDecomposition(A,&Q)</code></strong>
</pre><p> Vous obtenez la matrice supérieure dans une variable appelée <code class="varname">R</code> et la
matrice orthogonale (unitaire) dans <code class="varname">Q</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Renvoie le quotient de Rayleigh (aussi appelé le quotient ou
rapport de Rayleigh-Ritz) d'une matrice et d'un vecteur.</p><p>
@@ -146,35 +146,35 @@ superdiagonal being all ones. It is the Jordan block matrix of one zero eigenva
<a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</p></dd><dt><span
class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Alias : <code class="function">RotationMatrix</code></p><p>Renvoie la matrice correspondant à
la rotation centrée sur l'origine dans R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Renvoie la matrice correspondant à la rotation centrée sur l'origine dans R<sup>3</sup>
autour de l'axe des x.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre class="synopsis">Rotation3DY
(angle)</pre><p>Renvoie la matrice correspondant à la rotation ce
ntrée sur l'origine dans R<sup>3</sup> autour de l'axe des y.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Renvoie la matrice correspondant à la rotation centrée sur l'origine dans R<sup>3</sup>
autour de l'axe des z.</p></dd><dt><span class="term"><a
name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre class="synopsis">RowSpace (M)</pre><p>Renvoie
une matrice de base pour l'espace vectoriel engendré par les lignes d'une matrice.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Évalue (v, w) par rapport à la forme sesquilinéaire donnée
par la matrice A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Renvoie une fon
ction qui évalue deux vecteurs par rapport à la forme sesquilinéaire donnée par A.</p></dd><dt><span
class="term"><a name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returns the Smith normal form of a matrix over fields (will
end up with 1's on the diagonal).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,params...)</pre><p>Résout le système linéaire Mx=V, renvoie V s'il y
a une solution unique ou <code class="constant">null</code> sinon. Deux références d'arguments
supplémentaires peuvent être utilisés pour recevoir les réductions de M et V.</p></dd><dt><span
class="term"><a name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre
class="synopsis">ToeplitzMatrix (c, r...)</pre><p>Renvoie la matrice de Toeplitz construite à partir de la
première colonne c et (éventuellement) de la première ligne r. Si seule la colonne c est fournie alors elle
est conjuguée et la version non conjuguée est utilisée pour la première ligne pour fournir une matrice
hermitienne (si le premier élément est réel bien sûr).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Alias : <code class="function">trace</code></p><p>Calcule la trace d'une
matrice, c'est-à-dire la somme des éléments diagonaux.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Matrice transposée. C'est identique à l'opérateur <strong
class="userinput"><code>.'</code></strong></p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alias : <code class="function">vander</code></p><p>Renvoie la
matrice de Vandermonde.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>L'angle entre deux vecteurs par rapport au produit scalaire donné par <code
class="varname">B</code>. Si <code class="varname">B</code> n'est pas fourni alors le produit hermitien
standard est utilisé. <code class="varname">B</code> peut être soit une forme sesquilinéaire à deux arguments
soit une matrice fournissant une forme sesquilinéaire.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Somme directe des espaces vectoriels M et
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersection des sous-espaces donnés par M et
N.</p></dd><dt><span cl
ass="term"><a name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>Somme des espaces vectoriels M et N, c'est-à-dire {w |
w=m+n, m dans M, n dans N}.</p></dd><dt><span class="term"><a
name="gel-function-adj"></a>adj</span></dt><dd><pre class="synopsis">adj (m)</pre><p>Alias : <code
class="function">Adjugate</code></p><p>Renvoie la matrice adjointe d'une matrice.</p></dd><dt><span
class="term"><a name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Alias
: <code class="function">CREF</code><code class="function">ColumnReducedEchelonForm</code></p><p>Calcule la
forme échelonnée réduite en colonnes.</p></dd><dt><span class="term"><a
name="gel-function-det"></a>det</span></dt><dd><pre class="synopsis">det (M)</pre><p>Alias : <code
class="function">Determinant</code></p><p>Renvoie le déterminant d'une matrice.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Alias : <code class="function">REF</code><code
class="function">RowEchelonForm</code></p><p>Renvoie la matrice échelonnée en lignes (row echelon) d'une
matrice. C'est-à-dire effectue une élimination de Gauss de <code class="varname">M</code>. Les lignes de
pivot sont divisées pour que tous les pivots soient égaux à 1.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alias : <code class="function">RREF</code><code
class="function">ReducedRowEchelonForm</code></p><p>Renvoie la matrice échelonnée réduite en lignes (reduced
row echelon) d'une matrice. C'est-à-dire effectue une élimination de Gauss-Jordan de <code
class="varname">M</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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Combinatoire</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s10.html b/help/fr/html/ch11s10.html
index ae09b72..dfaa0a6 100644
--- a/help/fr/html/ch11s10.html
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title="Chapitre 11. Liste des fonctions GEL"><link rel="prev" href="ch11s09.html" title="Algèbre
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link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Combinatoire</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s09.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des
fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s11.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a
name="genius-gel-function-list-combinatorics"></a>Combinatoire</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Renvoie le
<code class="varname">n</code>-ième nombre catalan.</p><p>
See
<a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Renvoie toutes les combinaisons de k nombres de 1 à n comme un vecteur de vecteurs (consultez
aussi <a class="link"
href="ch11s10.html#gel-function-NextCombination">NextCombination</a>).</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorielle : <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Renvoie toutes les combinaisons de k nombres de 1 à n comme un vecteur de vecteurs (consultez
aussi <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorielle : <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
See
<a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Factorielle : <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
@@ -12,13 +15,36 @@
<a class="ulink" href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre
class="synopsis">Fibonacci (x)</pre><p>Alias : <code class="function">fib</code></p><p>Calcule le <code
class="varname">n</code>-ième nombre de Fibonacci. C'est-à-dire le nombre défini de manière récursive par
<strong class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</code></strong> et <strong
class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,param...)</pre><p>Calcule le nombre de Frobenius. C'est-à-dire calcule le plus petit nombre qui ne peut
pas être obtenu comme une combinaison linéaire d'entiers non négatifs d'un vecteur donné d'entiers non
négatifs. Le vecteur peut être fourni sous la forme de nombre séparés ou d'un seul vecteur. Tous les nombres
fournis doivent avoir un PGCD de 1.</p><p>Consultez <a class="ulink"
href="http://mathworld.wolfram.com/FrobeniusNumber.html"; target="_top">Mathworld</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(règle_de_combinaison)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</sp
an></dt><dd><pre class="synopsis">GreedyAlgorithm (n,v)</pre><p>Trouve le vecteur <code
class="varname">c</code> d'entiers non négatifs tel que le produit scalaire par <code
class="varname">v</code> est égal à n. Si ce n'est pas possible, renvoie <code class="constant">null</code>.
<code class="varname">v</code> doit être fourni trié dans l'ordre croissant et doit être composé d'entier non
négatif.</p><p>Consultez <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alias : <code class="function">HarmonicH</code></p><p>Nombre harmonique, le <code
class="varname">n</code>-ième nombre harmonique d'ordre <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="sy
nopsis">Hofstadter (n)</pre><p>Fonction de Hofstadter q(n) définie par q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (valeurs_ensemencement,règle_de_combinaison,n)</pre><p>Calcule la
relation de récurrence linéaire en utilisant l'algorithme de Galois.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,param...)</pre><p>Calcule les coefficients multinomiaux. Prend un vecteur de <code
class="varname">k</code> entiers non négatifs et calcule les coefficients multinomiaux. Cela correspond aux
coefficients dans le polynôme homogène à <code class="varname">k</code> variables avec les puissances
correspondantes.</p><p>La formule pour <strong class="userinput"><code>Multinomial(a,b,c)</code></strong>
peut s'écrire sous la forme�
�:</p><pre class="programlisting">(a+b+c)! / (a!b!c!)
+ </p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,param...)</pre><p>
+ Calculate the Frobenius number. That is calculate largest
+ number that cannot be given as a non-negative integer linear
+ combination of a given vector of non-negative integers.
+ The vector can be given as separate numbers or a single vector.
+ All the numbers given should have GCD of 1.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(règle_de_combinaison)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Trouve le vecteur <code class="varname">c</code> d'entiers non négatifs tel que le produit
scalaire par <code class="varname">v</code> est égal à n. Si ce n'est pas possible, renvoie <code
class="constant">null</code>. <code class="varname">v</code> doit être fourni trié dans l'ordre croissant et
doit être composé d'entier non négatif.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alias : <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Fonction de Hofstadter q(n) définie par q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (valeurs_ensemencement,règle_de_combinaison,n)</pre><p>Calcule la
relation de récurrence linéaire en utilisant l'algorithme de Galois.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,param...)</pre><p>Calcule les coefficients multinomiaux. Prend un vecteur de <code
class="varname">k</code> entiers non négatifs et calcule les coefficients multinomiaux. Cela correspond aux
coefficients dans le polynôme homogène à <code class="varname">k</code> variables avec les puissances
correspondantes.</p><p>La formule pour <strong class="userinput"><code>Multinomial(a,b,c)</code></strong>
peut s'écrire sous la forme :</p><pre class="programlisting">(a+b+c)! / (a!b!c!)
</pre><p> En d'autres termes, si vous n'avez que deux éléments alors <strong
class="userinput"><code>Multinomial(a,b)</code></strong> est la même chose que <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> ou <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Calcule la combinaison qui apparaîtrait après v dans un appel à la fonction combinations, la
première combinaison devrait être <strong class="userinput"><code>[1:k]</code></strong>. Cette fonction est
utile si vous devez parcourir beaucoup de combinaisons et que vous ne voulez pas gaspiller de la mémoire pour
les enregistrer.</p><p>
@@ -34,13 +60,20 @@ do (
) while not IsNull(n:=NextCombination(n,6));</code></strong>
</pre><p>
See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Get the Pascal's triangle as a matrix. This will return
an <code class="varname">i</code>+1 by <code class="varname">i</code>+1 lower diagonal
matrix that is the Pascal's triangle after <code class="varname">i</code>
iterations.</p><p>
See
<a class="ulink" href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> for
more information.
- </p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Renvoie toutes les permutations de <code class="varname">k</code> nombres de 1 à <code
class="varname">n</code> comme un vecteur de vecteurs.</p><p>Consultez <a class="ulink"
href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> ou <a class="ulink"
href="http://fr.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alias : <code class="function">Pochhammer</code></p><p>Factorielle croissante (Pochhammer) :
(n)_k = n(n+1)...(n+(k-1)).</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Renvoie toutes les permutations de <code class="varname">k</code> nombres de 1 à <code
class="varname">n</code> comme un vecteur de vecteurs.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alias : <code class="function">Pochhammer</code></p><p>Factorielle croissante (Pochhammer) :
(n)_k = n(n+1)...(n+(k-1)).</p><p>
See
<a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alias : <code
class="function">StirlingS1</code></p><p>Nombre de Stirling du premier type.</p><p>
@@ -58,4 +91,8 @@ do (
See
<a class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre
class="synopsis">nPr (n,r)</pre><p>Calculate the number of permutations of size
- <code class="varname">r</code> of numbers from 1 to <code
class="varname">n</code>.</p><p>Consultez <a class="ulink"
href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> ou <a class="ulink"
href="http://fr.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Précédent</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Niveau supérieur</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch11s11.html">Suivant</a></td></tr><tr><td width="40%" align="left"
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Analyse</td></tr></table></div></body></html>
+ <code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
+ See
+ <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Précédent</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Niveau supérieur</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch11s11.html">Suivant</a></td></tr><tr><td width="40%" align="left"
valign="top">Algèbre linéaire </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Sommaire</a></td><td width="40%" align="right" valign="top">
Analyse</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s11.html b/help/fr/html/ch11s11.html
index d3c2ba5..985412d 100644
--- a/help/fr/html/ch11s11.html
+++ b/help/fr/html/ch11s11.html
@@ -18,7 +18,11 @@ the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code>
the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, while
<strong class="userinput"><code>b@(n)</code></strong> refers to the term
<strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Either <code class="varname">a</code>
-or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>Consultez <a
class="ulink" href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a
class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(fonc,début,inc)</pre><p>Essaie de calculer un produit infini pour une fonction à un seul
argument.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (fonc,param,début,inc)</pre><p>Essaie de calculer un produit infini pour
une fonction à double arguments avec func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(fonc,début,inc)
</pre><p>Essaie de calculer une somme infinie pour une fonction à un seul argument.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (fonc,param,début,inc)</pre><p>Essaie de calculer une somme infinie pour une
fonction à double arguments avec func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Essaie de voir si une fonction à valeur réelle est continue en x0 en calculant la limite en ce
point.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Teste de différentiabilité en approchant les limites gauche
et droite et en les comparant.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit (f,x0)</pre><
p>Calcule la limite gauche d'une fonction à valeurs réelles en x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Calcule la
limite d'une fonction à valeur réelle en x0. Essaie de calculer les deux limites à gauche et à
droite.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Intégration par la méthode des rectangles.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alias : <code
class="function">NDerivative</code></p><p>Essaie de calculer la dérivée par méthode numérique.</p><p>
+or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(fonc,début,inc)</pre><p>Essaie de calculer un produit infini pour une fonction à un seul
argument.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (fonc,param,début,inc)</pre><p>Essaie de calculer un produit infini pour
une fonction à double arguments avec func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(fonc,début,inc)</pre><p>Essaie de calculer une somme infinie pour une fonction à un seul
argument.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre class="synopsis">InfiniteSum2
(fonc,param,début,inc)</pre><p>Essaie de calculer une somme infinie pour une fonction à doubl
e arguments avec func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Essaie de voir si une fonction à valeur réelle est continue en x0 en calculant la limite en ce
point.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Teste de différentiabilité en approchant les limites gauche
et droite et en les comparant.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calcule la limite gauche d'une fonction à valeurs réelles en x0.</p></dd><dt><span
class="term"><a name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit
(f,x0)</pre><p>Calcule la limite d'une fonction à valeur réelle en x0. Essaie de calculer les deux limites à
gauche et à droite.
</p></dd><dt><span class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre
class="synopsis">MidpointRule (f,a,b,n)</pre><p>Intégration par la méthode des rectangles.</p></dd><dt><span
class="term"><a name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alias : <code
class="function">NDerivative</code></p><p>Essaie de calculer la dérivée par méthode numérique.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Return a vector of vectors <strong
class="userinput"><code>[a,b]</code></strong>
@@ -29,13 +33,21 @@ the Fourier series of
on <strong class="userinput"><code>[-L,L]</code></strong> and extended periodically) with coefficients
up to <code class="varname">N</code>th harmonic computed numerically. The coefficients are
computed by numerical integration using
-<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
on <strong class="userinput"><code>[-L,L]</code></strong> and extended periodically) with coefficients
up to <code class="varname">N</code>th harmonic computed numerically. This is the
trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
-<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the cosine Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
@@ -46,14 +58,22 @@ computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.
Note that <strong class="userinput"><code>a@(1)</code></strong> is
the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
-the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
+the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
take the even periodic extension and compute the Fourier series, which
only has cosine terms. The series is computed up to the
<code class="varname">N</code>th harmonic. The coefficients are
computed by numerical integration using
-<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the sine Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
@@ -61,14 +81,22 @@ take the odd periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the
<code class="varname">N</code>th harmonic. The coefficients are
computed by numerical integration using
-<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
we take <code class="function">f</code> defined on <strong class="userinput"><code>[0,L]</code></strong>
take the odd periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the
<code class="varname">N</code>th harmonic. The coefficients are
computed by numerical integration using
-<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier"; target="_top">Wikipedia</a> ou <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> pour plus
d'informations.</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Intégration de f entre a et b, en suivant la règle définie
par <code class="varname">NumericalIntegralFunction</code> et en utilisant les <code
class="varname">NumericalIntegralSteps</code> pas.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Essaie de calculer la dérivée à gauche p
ar méthode numérique.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Essaie de
calculer la limite de f (step_fun(i)) lorsque i va de 1 à N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Essaie de calculer la dérivée à droite par méthode
numérique.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
+<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
+ </p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Intégration de f entre a et b, en suivant la règle définie
par <code class="varname">NumericalIntegralFunction</code> et en utilisant les <code
class="varname">NumericalIntegralSteps</code> pas.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Essaie de calculer la dérivée à gauche par méthode
numérique.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Essaie de
calculer la limite de f (step_fun(i)) lorsque i va de 1 à N.</p></dd><dt><span class="term"><a name
="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Essaie de calculer la dérivée à droite par méthode
numérique.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
<code class="function">f</code> with half period <code class="varname">L</code>. That
is a function defined on the interval <strong class="userinput"><code>[0,L]</code></strong>
extended to be odd on <strong class="userinput"><code>[-L,L]</code></strong> and then
diff --git a/help/fr/html/ch11s12.html b/help/fr/html/ch11s12.html
index 4ae3080..f7ce82d 100644
--- a/help/fr/html/ch11s12.html
+++ b/help/fr/html/ch11s12.html
@@ -1,21 +1,21 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Fonctions</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manuel
de Genius"><link rel="up" href="ch11.html" title="Chapitre 11. Liste des fonctions GEL"><link rel="prev"
href="ch11s11.html" title="Analyse"><link rel="next" href="ch11s13.html" title="Résolution
d'équations"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Fonctions</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des fonctions
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s13.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a name="genius-gel-function-list-functions"></a>Fonctions</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Alias :
<code class="function">Arg</code> <code class="function">arg</code></p><p>Renvoie l'argument (angle) d'un
nombre complexe.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0 (x)</pre><p>Bessel
function of the first kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Bessel
function of the first kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Bessel
function of the first kind of order <code class="varname">n</code>. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Bessel
function of the second kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Bessel
function of the second kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Bessel
function of the second kind of order <code class="varname">n</code>. Only implemented for real
numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Renvoie 1 si et seulement si tous les éléments sont nuls.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alias : <code class="function">erf</code></p><p>Fonction erreur, 2/sqrt(pi) * int_0^x e^(-t^2)
dt.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
@@ -26,7 +26,7 @@
</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alias : <code class="function">Gamma</code></p><p>La fonction Gamma. Seules les valeurs réelles
sont actuellement implémentées.</p><p>
See
<a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Renvoie 1 si et seulement si tous les éléments sont égaux.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>
The principal branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>.
@@ -37,7 +37,7 @@
See <a class="link" href="ch11s12.html#gel-function-LambertWm1"><code
class="function">LambertWm1</code></a> for the other real branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>
The minus-one branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>
@@ -47,29 +47,37 @@
See <a class="link" href="ch11s12.html#gel-function-LambertW"><code
class="function">LambertW</code></a> for the principal branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (fonc,x,incr)</pre><p>Cherche la première valeur pour laquelle
f(x)=0.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Transformation de Möbius du disque vers lui-même en faisant
correspondre a à 0.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Transformation de Möbius utilisant le rapport croisé en prenant z2, z3, z4 à 1, 0 et
l'infini respectivement.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Transformation de Möbius utilisant le rapport
croisé en prenant l'infini à l'infini et z2, z3 à 1 et 0 respectivement.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Transformation de Möbius utilisant le rapport
croisé en prenant l'infini à 1 et z3, z4 à 0 et l'infini respectivement.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Transformation de Möbius utilisant le rapport
croisé en prenant l'infini à 0 et z2, z4 à 1 et l'infini respectivement.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Le noyau de Poisson sur D(0,1) (non normalisé à 1, donc son intégrale vaut 2
pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Le noyau de Poisson sur D(0,R) (non normalisé à
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Alias : <code class="function">zeta</code></p><p>Fonction zeta de
Riemann (seules les valeurs réelles sont actuellement implémentées).</p><p>
See
<a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
- </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>La fonction échelon vaut 0 pour x<0, 1 sinon. C'est l'intégrale de
la fonction delta de Dirac. Elle est aussi appelée fonction d'Heaviside.</p><p>Consultez <a class="ulink"
href="http://fr.wikipedia.org/wiki/Fonction_de_Heaviside"; target="_top">Wikipedia</a> pour plus
d'informations.</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>La fonction <code class="function">cis</code> est la même que <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Convertit
les degrés en radians.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Convertit
les radians en degrés.</p></dd><dt><span class="term"><a
name="gel-function-sinc"></a>sinc</span></dt><dd><pre class="synopsis">sinc (x)</pre><p>Calculates the
unnormalized sinc function, that is
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>La fonction échelon vaut 0 pour x<0, 1 sinon. C'est l'intégrale de
la fonction delta de Dirac. Elle est aussi appelée fonction d'Heaviside.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>La fonction <code class="function">cis</code> est la même que <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Convertit
les degrés en radians.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Convertit
les radians en degrés.</p></dd><dt><span class="term"><a
name="gel-function-sinc"></a>sinc</span></dt><dd><pre class="synopsis">sinc (x)</pre><p>Calculates the
unnormalized sinc function, that is
<strong class="userinput"><code>sin(x)/x</code></strong>.
If you want the normalized function call <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
</p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Précédent</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Niveau supérieur</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Suivant</a></td></tr><tr><td width="40%" align="left" valign="top">Analyse </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Sommaire</a></td><td width="40%" align="right"
valign="top"> Résolution d'équations</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s13.html b/help/fr/html/ch11s13.html
index 2242fc6..4afbc59 100644
--- a/help/fr/html/ch11s13.html
+++ b/help/fr/html/ch11s13.html
@@ -2,7 +2,7 @@
See
<a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
@@ -15,12 +15,12 @@
</p><p>Les systèmes peuvent être résolus en ayant uniquement <code class="varname">y</code> sous la
forme d'un vecteur (colonne) partout. C'est-à-dire <code class="varname">y0</code> peut être un vecteur et
dans ce cas <code class="varname">f</code> doit prendre un nombre <code class="varname">x</code> et un
vecteur de la même taille comme deuxième argument et doit renvoyer un vecteur de la même taille.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
@@ -53,7 +53,7 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Find root of a function using the bisection method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
@@ -80,7 +80,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Find zeros using Newton's method. <code class="varname">f</code> is
the function and <code class="varname">df</code> is the derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
@@ -94,27 +94,28 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Calcule les racines d'un polynôme (de degré 1 à 4) en utilisant une des formules adaptée à ce
type de polynôme. Le polynôme doit être fourni sous la forme d'un vecteur de coefficients. Par exemple
<strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> correspond au vecteur <strong
class="userinput"><code>[1,2,0,4]</code></strong>. Renvoie un vecteur colonne contenant les
solutions.</p><p>La fonction appelle <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> et <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre class=
"synopsis">QuadraticFormula (p)</pre><p>Calcule les racines d'un polynôme quadratique (de degré 2) en
utilisant la formule quadratique. Le polynôme doit être fourni sous la forme d'un vecteur de coefficients.
<strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> correspond au vecteur <strong
class="userinput"><code>[1,2,3]</code></strong>. Renvoie un vecteur colonne contenant les deux
solutions.</p><p>
See
- <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Calcule les racines d'un polynôme quartique (de degré 4) en utilisant la formule quartique. Le
polynôme doit être fourni sous la forme d'un vecteur de coefficients. <strong class="userinput"><code>5*x^4 +
2*x + 1</code></strong> correspond au vecteur <strong class="userinput"><code>[1,2,0,0,5]</code></strong>.
Renvoie un vecteur colonne contenant les quatre solutions.</p><p>
See
<a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Utilise la méthode classique non adaptative de Runge-Kutta du quatrième ordre pour
résoudre numériquement y'=f(x,y) avec les valeurs initiales <code class="varname">x0</code>, <code
class="varname">y0</code> allant vers <code class="varname">x1</code> avec <code class="varname">n</code>
incréments, renvoie <code class="varname">y</code> en <code class="varname">x1</code>.</p><p>Les systèmes
peuvent être résolus en ayant uniquement <code class="varname">y</code> sous la forme d'un vecteur (colonne)
partout. C'est-à-dire <code class="varname">y0</code> peut être un vecteur et dans ce cas <code
class="varname">f</code> doit prendre un nombre <code class="varname">x</code> et un vecteur de la même
taille comme deuxième argument et doit renvoyer un vecteur de la même taille.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
@@ -142,5 +143,5 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Précédent</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Niveau supérieur</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">Suivant</a></td></tr><tr><td width="40%" align="left" valign="top">Fonctions </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Sommaire</a></td><td width="40%" align="right"
valign="top"> Statistiques</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s14.html b/help/fr/html/ch11s14.html
index 33d53b3..0bc6bcf 100644
--- a/help/fr/html/ch11s14.html
+++ b/help/fr/html/ch11s14.html
@@ -1 +1,26 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistiques</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manuel de Genius"><link rel="up" href="ch11.html"
title="Chapitre 11. Liste des fonctions GEL"><link rel="prev" href="ch11s13.html" title="Résolution
d'équations"><link rel="next" href="ch11s15.html" title="Polynômes"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Statistiques</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des
fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class=
"title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Statistiques</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alias :
<code class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Calcule la moyenne de toute une matrice.</p><p>Consultez <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> pour plus
d'informations.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Intégrale de la fonction de Gauss de 0 à <code
class="varname">x</code> (aire sous la courbe normale).</p><p>Consultez <a class="ulink"
href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> pour
plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Fonction distribution de Gauss normalisée (courbe normale).</p><p>Consultez <a
class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> pour
plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-Median"></a>Median</span></dt><dd><pre class="synopsis">Median (m)</pre><p>Alias : <code
class="function">median</code></p><p>Calcule la médiane de toute une matrice.</p><p>Consultez <a
class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> pour
plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alias : <code class="function">stdevp</code>
</p><p>Calcule l'écart type de la population de toute une matrice.</p></dd><dt><span class="term"><a
name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre class="synopsis">RowAverage
(m)</pre><p>Alias : <code class="function">RowMean</code></p><p>Calcule la moyenne de chaque ligne d'une
matrice.</p><p>Consultez <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> pour plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre class="synopsis">RowMedian
(m)</pre><p>Calcule la médiane de chaque ligne d'une matrice et renvoie un vecteur colonne.</p><p>Consultez
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> pour
plus d'informations.</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationS
tandardDeviation (m)</pre><p>Alias : <code class="function">rowstdevp</code></p><p>Calcule l'écart type de
la population des lignes d'une matrice et renvoie un vecteur colonne.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alias : <code
class="function">rowstdev</code></p><p>Calcule l'écart type des lignes d'une matrice et renvoie un vecteur
colonne.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alias : <code class="function">stdev</code></p><p>Calcule
l'écart type de toute une matrice.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Précédent</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.h
tml">Niveau supérieur</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Suivant</a></td></tr><tr><td width="40%" align="left" valign="top">Résolution d'équations
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Sommaire</a></td><td width="40%"
align="right" valign="top"> Polynômes</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Statistiques</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manuel de Genius"><link rel="up" href="ch11.html"
title="Chapitre 11. Liste des fonctions GEL"><link rel="prev" href="ch11s13.html" title="Résolution
d'équations"><link rel="next" href="ch11s15.html" title="Polynômes"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Statistiques</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des
fonctions GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class=
"title" style="clear: both"><a
name="genius-gel-function-list-statistics"></a>Statistiques</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alias :
<code class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Intégrale de la fonction de Gauss de 0 à <code
class="varname">x</code> (aire sous la courbe normale).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Fonction distribution de Gauss normalisée (courbe normale).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Alias : <code class="function">median</code></p><p>Calcule la médiane de
toute une matrice.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alias : <code
class="function">stdevp</code></p><p>Calcule l'écart type de la population de toute une
matrice.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Alias : <code class="function">RowMean</code></p><p>Calculate average
of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calcule la médiane de chaque ligne d'une matrice et renvoie un vecteur
colonne.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alias : <code
class="function">rowstdevp</code></p><p>Calcule l'écart type de la population des lignes d'une matrice et
renvoie un vecteur colonne.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alias : <code
class="function">rowstdev</code></p><p>Calcule l'écart type des lignes d'une matrice et renvoie un vecteur
colonne.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alias : <code class="function">stdev</code></p><p>Calcule
l'écart type de toute une matrice.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summa
ry="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s13.html">Précédent</a>
</td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Niveau supérieur</a></td><td
width="40%" align="right"> <a accesskey="n" href="ch11s15.html">Suivant</a></td></tr><tr><td width="40%"
align="left" valign="top">Résolution d'équations </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Sommaire</a></td><td width="40%" align="right" valign="top">
Polynômes</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s15.html b/help/fr/html/ch11s15.html
index e213162..78097a5 100644
--- a/help/fr/html/ch11s15.html
+++ b/help/fr/html/ch11s15.html
@@ -13,5 +13,5 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Renvoie la dérivée seconde du polynôme (comme
vecteur).</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Prend la dérivée du polynôme (comme vecteur).</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Fabrique une fonction à partir d'un polynôme (comme vecteur).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Fabrique une chaîne à partir d'un polynôme (comme vecteur).</p></dd><dt><span
class="term"><a name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre
class="synopsis">SubtractPol
y (p1,p2)</pre><p>Soustrait deux polynômes (comme vecteur).</p></dd><dt><span class="term"><a
name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly (p)</pre><p>Tronque
les zéros d'un polynôme (défini comme vecteur).</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s14.html">Précédent</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Niveau supérieur</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Suivant</a></td></tr><tr><td width="40%" align="left" valign="top">Statistiques </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Sommaire</a></td><td width="40%" align="right"
valign="top"> Théorie des ensembles</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s18.html b/help/fr/html/ch11s18.html
index 124f411..f9d07be 100644
--- a/help/fr/html/ch11s18.html
+++ b/help/fr/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Divers</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manuel
de Genius"><link rel="up" href="ch11.html" title="Chapitre 11. Liste des fonctions GEL"><link rel="prev"
href="ch11s17.html" title="Commutative Algebra"><link rel="next" href="ch11s19.html" title="Calcul
symbolique"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Divers</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des fonctions
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-gel-function-list-miscellaneous"></a>Divers</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convertit un vecteur de valeurs ASCII en chaîne.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convertit un vecteur d'indices en chaîne de
caractères. Les indices correspondent à la position dans la chaîne <code class="literal">alphabet</code>, en
commençant à zéro.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(chaîne)</pre><p>Convertit une chaîne en vecteur de valeurs ASCII.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet
</span></dt><dd><pre class="synopsis">StringToAlphabet (chaîne,alphabet)</pre><p>Convertit une <code
class="literal">chaîne</code> de caractères en un vecteur d'indices correspondant à la position dans la
chaîne <code class="literal">alphabet</code> (en commençant à zéro). Lorsque le caractère n'est pas dans
l'<code class="literal">alphabet</code>, l'indice est -1.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s17.html">Précédent</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Niveau supérieur</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Suivant</a></td></tr><tr><td width="40%" align="left" valign="top">Commutative Algebra
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Sommaire</a></td><td width="40%"
align="right" valign="top"> Calcul symbolique</td></tr></table></di
v></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Divers</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manuel
de Genius"><link rel="up" href="ch11.html" title="Chapitre 11. Liste des fonctions GEL"><link rel="prev"
href="ch11s17.html" title="Commutative Algebra"><link rel="next" href="ch11s19.html" title="Calcul
symbolique"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Divers</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Précédent</a> </td><th width="60%" align="center">Chapitre 11. Liste des fonctions
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Suivant</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a name="genius-gel-function-list-miscellaneous"></a>Divers</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(chaîne)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (chaîne,alphabet)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Précédent</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Niveau supérieur</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch11s19.html">Suivant</a></td></tr><tr><td width="40%" align="left"
valign="top">Commutative Algebra </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Sommaire</a></td><td width="40%" align="right" valign="top"> Calcul
symbolique</td></tr></table></div></body></html>
diff --git a/help/fr/html/ch11s20.html b/help/fr/html/ch11s20.html
index 2cd34e4..df2cc6c 100644
--- a/help/fr/html/ch11s20.html
+++ b/help/fr/html/ch11s20.html
@@ -83,7 +83,7 @@
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
</pre><p>
@@ -134,7 +134,7 @@
Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
</pre><p>
@@ -273,7 +273,7 @@
<code class="varname">n</code> by 3 matrix for a longer polyline.
</p><p>
Extra parameters can be added to specify line color, thickness,
- arrows, the plotting window, or legend.
+ the plotting window, or legend.
You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
<strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>,
diff --git a/help/fr/html/index.html b/help/fr/html/index.html
index 621e46e..9e9d3af 100644
--- a/help/fr/html/index.html
+++ b/help/fr/html/index.html
@@ -1,5 +1,5 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Manuel de
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Manuel de l'Outil de maths Genius."><link rel="home" href="index.html" title="Manuel de
Genius"><link rel="next" href="ch01.html" title="Chapitre 1. Introduction"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Manuel de Genius</th></tr><tr><td width="20%"
align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Suivant</a></td></tr></table><hr></div><div lang="fr" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Manuel de Genius</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><span class="fir
stname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Oklahoma State University<br></span><div class="address"><p> <code class="email"><<a
class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code> </p></div></div></div><div
class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">Université de Queensland,
Australie<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:kaiw itee
uq edu au">kaiw itee uq edu au</a>></code> </p></div></div></div></div></div><div><p
class="releaseinfo">This manual describes version 1.0.22 of Genius.
- </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2010-11 Bruno
Brouard (annoa b gmail com)</p></div><div><p class="copyright">Copyright © 2011 Luc Pionchon (pionchon luc
gmail com)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Permission vous est donnée de
copier, distribuer et/ou modifier ce document selon les termes de la Licence GNU Free Documentation License,
Version 1.1 ou ultérieure publiée par la Free Software Foundation sans section inaltérable, sans texte de
première page de couverture ni texte de dernière page de couverture. Vous trouverez un exemplaire de cette
licence en suivant ce <a class="ulink" href="ghelp:fdl" target="_top">lien</a> ou dans le fichier
COPYING-DOCS fourni avec le présent manuel.</p><p>Ce manuel fait partie de la collection de manuels GNOME
distribués selon les term
es de la licence de documentation libre GNU. Si vous souhaitez distribuer ce manuel indépendamment de la
collection, vous devez joindre un exemplaire de la licence au document, comme indiqué dans la section 6 de
celle-ci.</p><p>La plupart des noms utilisés par les entreprises pour distinguer leurs produits et services
sont des marques déposées. Lorsque ces noms apparaissent dans la documentation GNOME et que les membres du
projet de Documentation GNOME sont informés de l'existence de ces marques déposées, soit ces noms entiers,
soit leur première lettre est en majuscule.</p><p>LE PRÉSENT DOCUMENT ET SES VERSIONS MODIFIÉES SONT FOURNIS
SELON LES TERMES DE LA LICENCE DE DOCUMENTATION LIBRE GNU SACHANT QUE : </p><div class="orderedlist"><ol
class="orderedlist" type="1"><li class="listitem"><p>LE PRÉSENT DOCUMENT EST FOURNI « TEL QUEL », SANS AUCUNE
GARANTIE, EXPRESSE OU IMPLICITE, Y COMPRIS, ET SANS LIMITATION, LES GARANTIES DE MARCHANDABILITÉ,
D'ADÉQUATION �
� UN OBJECTIF PARTICULIER OU DE NON INFRACTION DU DOCUMENT OU DE SA VERSION MODIFIÉE. L'UTILISATEUR ASSUME
TOUT RISQUE RELATIF À LA QUALITÉ, À LA PERTINENCE ET À LA PERFORMANCE DU DOCUMENT OU DE SA VERSION DE MISE À
JOUR. SI LE DOCUMENT OU SA VERSION MODIFIÉE S'AVÉRAIT DÉFECTUEUSE, L'UTILISATEUR (ET NON LE RÉDACTEUR
INITIAL, L'AUTEUR, NI TOUT AUTRE PARTICIPANT) ENDOSSERA LES COÛTS DE TOUTE INTERVENTION, RÉPARATION OU
CORRECTION NÉCESSAIRE. CETTE DÉNÉGATION DE RESPONSABILITÉ CONSTITUE UNE PARTIE ESSENTIELLE DE CETTE LICENCE.
AUCUNE UTILISATION DE CE DOCUMENT OU DE SA VERSION MODIFIÉE N'EST AUTORISÉE AUX TERMES DU PRÉSENT ACCORD,
EXCEPTÉ SOUS CETTE DÉNÉGATION DE RESPONSABILITÉ ; </p></li><li class="listitem"><p>EN AUCUNE CIRCONSTANCE ET
SOUS AUCUNE INTERPRÉTATION DE LA LOI, QU'IL S'AGISSE D'UN DÉLIT CIVIL (Y COMPRIS LA NÉGLIGENCE), CONTRACTUEL
OU AUTRE, L'AUTEUR, LE RÉDACTEUR INITIAL, TOUT PARTICIPANT OU TOUT DISTRIBUTEUR DE CE DOCUMENT OU DE SA V
ERSION MODIFIÉE, OU TOUT FOURNISSEUR DE L'UNE DE CES PARTIES NE POURRA ÊTRE TENU RESPONSABLE À L'ÉGARD DE
QUICONQUE POUR TOUT DOMMAGE DIRECT, INDIRECT, PARTICULIER, OU ACCIDENTEL DE TOUT TYPE Y COMPRIS, SANS
LIMITATION, LES DOMMAGES LIÉS À LA PERTE DE CLIENTÈLE, À UN ARRÊT DE TRAVAIL, À UNE DÉFAILLANCE OU UN MAUVAIS
FONCTIONNEMENT INFORMATIQUE, OU À TOUT AUTRE DOMMAGE OU PERTE LIÉE À L'UTILISATION DU DOCUMENT ET DE SES
VERSIONS MODIFIÉES, MÊME SI LADITE PARTIE A ÉTÉ INFORMÉE DE L'ÉVENTUALITÉ DE TELS
DOMMAGES.</p></li></ol></div></div></div><div><div class="legalnotice"><a name="idm46125091179904"></a><p
class="legalnotice-title"><b>Votre avis</b></p><p>
+ </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2010-11 Bruno
Brouard (annoa b gmail com)</p></div><div><p class="copyright">Copyright © 2011 Luc Pionchon (pionchon luc
gmail com)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Permission vous est donnée de
copier, distribuer et/ou modifier ce document selon les termes de la Licence GNU Free Documentation License,
Version 1.1 ou ultérieure publiée par la Free Software Foundation sans section inaltérable, sans texte de
première page de couverture ni texte de dernière page de couverture. Vous trouverez un exemplaire de cette
licence en suivant ce <a class="ulink" href="ghelp:fdl" target="_top">lien</a> ou dans le fichier
COPYING-DOCS fourni avec le présent manuel.</p><p>Ce manuel fait partie de la collection de manuels GNOME
distribués selon les term
es de la licence de documentation libre GNU. Si vous souhaitez distribuer ce manuel indépendamment de la
collection, vous devez joindre un exemplaire de la licence au document, comme indiqué dans la section 6 de
celle-ci.</p><p>La plupart des noms utilisés par les entreprises pour distinguer leurs produits et services
sont des marques déposées. Lorsque ces noms apparaissent dans la documentation GNOME et que les membres du
projet de Documentation GNOME sont informés de l'existence de ces marques déposées, soit ces noms entiers,
soit leur première lettre est en majuscule.</p><p>LE PRÉSENT DOCUMENT ET SES VERSIONS MODIFIÉES SONT FOURNIS
SELON LES TERMES DE LA LICENCE DE DOCUMENTATION LIBRE GNU SACHANT QUE : </p><div class="orderedlist"><ol
class="orderedlist" type="1"><li class="listitem"><p>LE PRÉSENT DOCUMENT EST FOURNI « TEL QUEL », SANS AUCUNE
GARANTIE, EXPRESSE OU IMPLICITE, Y COMPRIS, ET SANS LIMITATION, LES GARANTIES DE MARCHANDABILITÉ,
D'ADÉQUATION �
� UN OBJECTIF PARTICULIER OU DE NON INFRACTION DU DOCUMENT OU DE SA VERSION MODIFIÉE. L'UTILISATEUR ASSUME
TOUT RISQUE RELATIF À LA QUALITÉ, À LA PERTINENCE ET À LA PERFORMANCE DU DOCUMENT OU DE SA VERSION DE MISE À
JOUR. SI LE DOCUMENT OU SA VERSION MODIFIÉE S'AVÉRAIT DÉFECTUEUSE, L'UTILISATEUR (ET NON LE RÉDACTEUR
INITIAL, L'AUTEUR, NI TOUT AUTRE PARTICIPANT) ENDOSSERA LES COÛTS DE TOUTE INTERVENTION, RÉPARATION OU
CORRECTION NÉCESSAIRE. CETTE DÉNÉGATION DE RESPONSABILITÉ CONSTITUE UNE PARTIE ESSENTIELLE DE CETTE LICENCE.
AUCUNE UTILISATION DE CE DOCUMENT OU DE SA VERSION MODIFIÉE N'EST AUTORISÉE AUX TERMES DU PRÉSENT ACCORD,
EXCEPTÉ SOUS CETTE DÉNÉGATION DE RESPONSABILITÉ ; </p></li><li class="listitem"><p>EN AUCUNE CIRCONSTANCE ET
SOUS AUCUNE INTERPRÉTATION DE LA LOI, QU'IL S'AGISSE D'UN DÉLIT CIVIL (Y COMPRIS LA NÉGLIGENCE), CONTRACTUEL
OU AUTRE, L'AUTEUR, LE RÉDACTEUR INITIAL, TOUT PARTICIPANT OU TOUT DISTRIBUTEUR DE CE DOCUMENT OU DE SA V
ERSION MODIFIÉE, OU TOUT FOURNISSEUR DE L'UNE DE CES PARTIES NE POURRA ÊTRE TENU RESPONSABLE À L'ÉGARD DE
QUICONQUE POUR TOUT DOMMAGE DIRECT, INDIRECT, PARTICULIER, OU ACCIDENTEL DE TOUT TYPE Y COMPRIS, SANS
LIMITATION, LES DOMMAGES LIÉS À LA PERTE DE CLIENTÈLE, À UN ARRÊT DE TRAVAIL, À UNE DÉFAILLANCE OU UN MAUVAIS
FONCTIONNEMENT INFORMATIQUE, OU À TOUT AUTRE DOMMAGE OU PERTE LIÉE À L'UTILISATION DU DOCUMENT ET DE SES
VERSIONS MODIFIÉES, MÊME SI LADITE PARTIE A ÉTÉ INFORMÉE DE L'ÉVENTUALITÉ DE TELS
DOMMAGES.</p></li></ol></div></div></div><div><div class="legalnotice"><a name="idm54"></a><p
class="legalnotice-title"><b>Votre avis</b></p><p>
To report a bug or make a suggestion regarding the <span class="application">Genius Mathematics
Tool</span>
application or this manual, please visit the
<a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius
diff --git a/help/make-makefile-am.sh b/help/make-makefile-am.sh
index c42eae6..fb279a5 100755
--- a/help/make-makefile-am.sh
+++ b/help/make-makefile-am.sh
@@ -1,6 +1,16 @@
#!/bin/sh
+# Edit this to add a language, then run this script
LANGS="cs de el es fr pt_BR ru sv"
+echo Creating Makefile.am...
+
+cat > Makefile.am <<EOF
+###################################################################
+# THIS FILE IS AUTOGENERATED DO NOT EDIT. EDIT make-makefile-am.sh
+###################################################################
+
+EOF
+
FIGUREBASENAMES=""
CFIGURES=""
for n in `ls C/figures/*.png 2>/dev/null` ; do
@@ -8,9 +18,35 @@ for n in `ls C/figures/*.png 2>/dev/null` ; do
FIGUREBASENAMES="$FIGUREBASENAMES `basename $n`"
done
-EXTRAFILES="$CFIGURES"
+CHTMLS=`ls C/html/*.html 2>/dev/null`
+CHTMLS=`echo $CHTMLS`
+EXTRAFILES="$CHTMLS $CFIGURES C/genius.xml C/legal.xml"
+
+CHTMLS=`echo $CHTMLS | sed 's/ / \\\\\n /g'`
+echo C
+
+cat >> Makefile.am <<EOF
+
+#########################################################
+#C
+
+THE_CFIGURES = $CFIGURES
+
+manualxmlCdir = \$(datadir)/genius/help/C
+manualxmlC_DATA = C/genius.xml C/legal.xml
+manualxmlCfiguresdir = \$(datadir)/genius/help/C/figures
+manualxmlCfigures_DATA = \$(THE_CFIGURES)
+
+manualhtmlCdir = \$(datadir)/genius/help/C/html
+manualhtmlC_DATA = $CHTMLS
+manualhtmlCfiguresdir = \$(datadir)/genius/help/C/html/figures
+manualhtmlCfigures_DATA = \$(THE_CFIGURES)
+
+
+EOF
for lang in $LANGS ; do
+ echo $lang
LANGBASENAMES=""
for n in `ls $lang/figures/*.png 2>/dev/null` ; do
LANGBASENAMES="$LANGBASENAMES `basename $n`"
@@ -20,8 +56,6 @@ for lang in $LANGS ; do
THEFIGURES=""
- echo
- echo $lang
for b in $LANGBASENAMES ; do
if test -e $lang/figures/$b ; then
@@ -32,20 +66,45 @@ for lang in $LANGS ; do
fi
done
- echo "$THEFIGURES"
-done
+ LANGHTMLS=`ls $lang/html/*.html 2>/dev/null`
+ LANGHTMLS=`echo $LANGHTMLS`
+ EXTRAFILES="$EXTRAFILES $LANGHTMLS $lang/genius.xml"
+ LANGHTMLS=`echo $LANGHTMLS | sed 's/ / \\\\\n /g'`
-echo
-echo EXTRA: $EXTRAFILES
+cat >> Makefile.am <<EOF
+#########################################################
+#$lang
+
+THE_${lang}FIGURES = $THEFIGURES
+
+manualxml${lang}dir = \$(datadir)/genius/help/${lang}
+manualxml${lang}_DATA = ${lang}/genius.xml
+manualxml${lang}figuresdir = \$(datadir)/genius/help/${lang}/figures
+manualxml${lang}figures_DATA = \$(THE_${lang}FIGURES)
+
+manualhtml${lang}dir = \$(datadir)/genius/help/${lang}/html
+manualhtml${lang}_DATA = $LANGHTMLS
+manualhtml${lang}figuresdir = \$(datadir)/genius/help/${lang}/html/figures
+manualhtml${lang}figures_DATA = \$(THE_${lang}FIGURES)
-cat > foo.out <<EOF
-###################################################################
-# THIS FILE IS AUTOGENERATED DO NOT EDIT. EDIT make-makefile-am.sh
-###################################################################
+EOF
+done
+
+EXTRAFILES=`echo genius.txt $EXTRAFILES | sed 's/ / \\\\\n /g'`
+cat >> Makefile.am <<EOF
+
+#########################################################
# Text version of the manual
+
manualdir = \$(datadir)/genius
manual_DATA = genius.txt
+#########################################################
+# Aaaaand here's all the files ...
+
+
EXTRA_DIST = $EXTRAFILES
EOF
+
+echo done
diff --git a/help/pt_BR/html/ch05s07.html b/help/pt_BR/html/ch05s07.html
index edce2a1..adb83f2 100644
--- a/help/pt_BR/html/ch05s07.html
+++ b/help/pt_BR/html/ch05s07.html
@@ -63,10 +63,12 @@ returns 3.
Element by element back division.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>
The mod operator. This does not turn on the <a class="link" href="ch05s06.html" title="Modular
Evaluation">modular mode</a>, but
- just returns the remainder of <strong class="userinput"><code>a/b</code></strong>.
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
- Element by element the mod operator. Returns the remainder
- after element by element integer <strong class="userinput"><code>a./b</code></strong>.
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
</p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>
Modular evaluation operator. The expression <code class="varname">a</code>
is evaluated modulo <code class="varname">b</code>. See <a class="xref" href="ch05s06.html"
title="Modular Evaluation">“Modular Evaluation”</a>.
@@ -102,21 +104,21 @@ returns 3.
greater than or equal to
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
- (can also be combine with the greater than operator).
+ (and they can also be combined with the greater than operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
Less than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
less than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
- (can also be combine with the less than or equal to operator).
+ (they can also be combined with the less than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
Greater than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
greater than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
- (can also be combine with the greater than or equal to operator).
+ (they can also be combined with the greater than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>
Comparison operator. If <code class="varname">a</code> is equal to
<code class="varname">b</code> it returns 0, if <code class="varname">a</code> is less
@@ -136,12 +138,12 @@ returns 3.
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
Logical xor.
- Returns true exactly one of
+ Returns true if exactly one of
<code class="varname">a</code> or <code class="varname">b</code> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
- Logical not. Returns the logical negation of <code class="varname">a</code>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>
@@ -160,7 +162,7 @@ returns 3.
Get element of a matrix in row <code class="varname">b</code> and column
<code class="varname">c</code>. If <code class="varname">b</code>,
<code class="varname">c</code> are vectors, then this gets the corresponding
- rows columns or submatrices.
+ rows, columns or submatrices.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>
Get row of a matrix (or multiple rows if <code class="varname">b</code> is a vector).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>
@@ -207,8 +209,8 @@ returns 3.
point numbers and is ever so slightly more precise than
<strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
- Make a imaginary number (multiply <code class="varname">a</code> by the
- imaginary). Note that normally the number <code class="varname">i</code> is
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
</p><pre class="programlisting">(a)*1i
</pre><p>
diff --git a/help/pt_BR/html/ch06s05.html b/help/pt_BR/html/ch06s05.html
index 7ffef75..35a2774 100644
--- a/help/pt_BR/html/ch06s05.html
+++ b/help/pt_BR/html/ch06s05.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Global Variables and
Scope of Variables</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual do Genius"><link rel="up" href="ch06.html" title="Capítulo 6. Programming
with GEL"><link rel="prev" href="ch06s04.html" title="Comparison Operators"><link rel="next"
href="ch06s06.html" title="Parameter variables"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Global Variables and Scope of Variables</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Anterior</a> </td><th width="60%"
align="center">Capítulo 6. Programming with GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Próxima</a></td></tr></table><hr></div><div class="sec
t1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Global Variables and Scope of Variables</h2></div></div></div><p>
GEL is a
- <a class="ulink" href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
dynamically scoped language</a>. We will explain what this
means below. That is, normal variables and functions are dynamically
scoped. The exception are
diff --git a/help/pt_BR/html/ch07s02.html b/help/pt_BR/html/ch07s02.html
index 6157ab4..c4285e7 100644
--- a/help/pt_BR/html/ch07s02.html
+++ b/help/pt_BR/html/ch07s02.html
@@ -3,10 +3,32 @@
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
- as just sequence of lines as if were entered on the command line.
+ as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
- parentheses).
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
</p><p>
The following code will produce an error when entered on the top
level of a program, while it will work just fine in a function.
diff --git a/help/pt_BR/html/ch11s04.html b/help/pt_BR/html/ch11s04.html
index 0b2903a..1ed036b 100644
--- a/help/pt_BR/html/ch11s04.html
+++ b/help/pt_BR/html/ch11s04.html
@@ -2,26 +2,26 @@
Catalan's Constant, approximately 0.915... It is defined to be the series where terms are
<strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, where <code class="varname">k</code>
ranges from 0 to infinity.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Aliases: <code class="function">gamma</code></p><p>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>The
Golden Ratio.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
round and uniform.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
is the exponential function
@@ -30,7 +30,7 @@
several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>
@@ -38,7 +38,7 @@
to its diameter. This is approximately 3.14159265359...
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Acima</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s05.html">Próxima</a></td></tr><tr><td width="40%" align="left"
valign="top">Parâmetros </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Principal</a></td><td width="40%" align="right" valign="top">
Numérico</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s05.html b/help/pt_BR/html/ch11s05.html
index fc6a047..53a6a91 100644
--- a/help/pt_BR/html/ch11s05.html
+++ b/help/pt_BR/html/ch11s05.html
@@ -5,7 +5,7 @@
to <strong class="userinput"><code>|x|</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
<a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
<a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
@@ -14,16 +14,16 @@ for more information.
</p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Replace very small number with zero.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Aliases: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calculates the complex conjugate of the complex number <code
class="varname">z</code>. If <code class="varname">z</code> is a vector or matrix,
all its elements are conjugated.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Get the denominator of a rational number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Return the fractional part of a number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Aliases: <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Division without remainder.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
<strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
<strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Check if argument is a possibly complex rational number.
That is, if both real and imaginary parts are
@@ -32,10 +32,10 @@ all its elements are conjugated.</p><p>
are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Check if argument is an integer (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Aliases: <code
class="function">IsNaturalNumber</code></p><p>Check if argument is a positive real integer. Note that
we accept the convention that 0 is not a natural number.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Check if argument is a rational number (non-complex). Of course rational simply means "not
stored as a floating point number."</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (num)</pre><p>Check if
argument is a real number.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Get
the numerator of a rational number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Aliases: <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Aliases: <code class="function">sign</code></p><p>Return the sign of a
number. That is returns
<code class="literal">-1</code> if value is negative,
<code class="literal">0</code> if value is zero and
@@ -61,12 +61,12 @@ value then <code class="function">Sign</code> returns the direction or 0.
logarithm</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Make number a floating point value. That is returns the floating point
representation of the number <code class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Aliases: <code
class="function">Floor</code></p><p>Get the highest integer less than or equal to <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>The natural logarithm, the logarithm to base <code
class="varname">e</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logarithm of <code
class="varname">x</code> base <code class="varname">b</code> (calls <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> if in modulo
mode), if base is not given, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> is used.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logarithm of <code
class="varname">x</code> base 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Aliases: <code
class="function">lg</code></p><p>Logarithm of <code class="varname">x</code> base 2.</p></dd><dt><span
class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre class="synop
sis">max (a,args...)</pre><p>Aliases: <code class="function">Max</code> <code
class="function">Maximum</code></p><p>Returns the maximum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,args...)</pre><p>Aliases: <code class="function">Min</code> <code
class="function">Minimum</code></p><p>Returns the minimum of arguments or matrix.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(size...)</pre><p>Generate random float in the range <code class="literal">[0,1)</code>.
diff --git a/help/pt_BR/html/ch11s06.html b/help/pt_BR/html/ch11s06.html
index 2ad026f..9880087 100644
--- a/help/pt_BR/html/ch11s06.html
+++ b/help/pt_BR/html/ch11s06.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometria</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual do Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. List of GEL functions"><link rel="prev" href="ch11s05.html" title="Numérico"><link
rel="next" href="ch11s07.html" title="Teoria dos números"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometria</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" styl
e="clear: both"><a name="genius-gel-function-list-trigonometry"></a>Trigonometria</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Aliases: <code
class="function">arccos</code></p><p>The arccos (inverse cos) function.</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Aliases: <code
class="function">arccosh</code></p><p>The arccosh (inverse cosh) function.</p></dd><dt><span class="term"><a
name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Aliases: <code
class="function">arccot</code></p><p>The arccot (inverse cot) function.</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Aliases: <code
class="function">arccoth</code></p><p>The arccoth (inverse coth) func
tion.</p></dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre
class="synopsis">acsc (x)</pre><p>Aliases: <code class="function">arccsc</code></p><p>The inverse cosecant
function.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Aliases: <code class="function">arccsch</code></p><p>The inverse
hyperbolic cosecant function.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Aliases: <code
class="function">arcsec</code></p><p>The inverse secant function.</p></dd><dt><span class="term"><a
name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech (x)</pre><p>Aliases: <code
class="function">arcsech</code></p><p>The inverse hyperbolic secant function.</p></dd><dt><span
class="term"><a name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin
(x)</pre><p>Aliases: <code
class="function">arcsin</code></p><p>The arcsin (inverse sin) function.</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Aliases: <code
class="function">arcsinh</code></p><p>The arcsinh (inverse sinh) function.</p></dd><dt><span class="term"><a
name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Aliases: <code
class="function">arctan</code></p><p>Calculates the arctan (inverse tan) function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Aliases: <code class="function">arctanh</code></p><p>The arctanh (inverse
tanh) function.</p></dd><dt><span class="term"><a name="gel-function-atan2"></a>atan2</span></dt><dd><pre
class="synopsis">atan2 (y, x)</pre><p>Aliases: <code class="function">arctan2</code></p><p>Calculates the
arctan2 function. If
<strong class="userinput"><code>x>0</code></strong> then it returns
@@ -11,11 +11,11 @@
rather than failing.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Calculates the cosine function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Calculates the hyperbolic cosine function.</p><p>
See
@@ -23,7 +23,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>The cotangent function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>The hyperbolic cotangent function.</p><p>
See
@@ -31,7 +31,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>The cosecant function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>The hyperbolic cosecant function.</p><p>
See
@@ -39,7 +39,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>The secant function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>The hyperbolic secant function.</p><p>
See
@@ -47,7 +47,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Calculates the sine function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Calculates the hyperbolic sine function.</p><p>
See
@@ -55,7 +55,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Calculates the tan function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>The hyperbolic tangent function.</p><p>
See
diff --git a/help/pt_BR/html/ch11s07.html b/help/pt_BR/html/ch11s07.html
index b7c17af..b7019f3 100644
--- a/help/pt_BR/html/ch11s07.html
+++ b/help/pt_BR/html/ch11s07.html
@@ -8,14 +8,14 @@
<a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Return the <code class="varname">n</code>th Bernoulli number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Aliases: <code class="function">CRT</code></p><p>Find the
<code class="varname">x</code> that solves the system given by
the vector <code class="varname">a</code> and modulo the elements of
<code class="varname">m</code>, using the Chinese Remainder Theorem.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Given two factorizations, give the factorization of the
@@ -23,7 +23,7 @@
F<sub>q</sub>, the finite field of order <code class="varname">q</code>, where <code
class="varname">q</code>
is a prime, using the Silver-Pohlig-Hellman algorithm.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Checks divisibility (if <code class="varname">m</code> divides <code
class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>
@@ -32,7 +32,7 @@
relatively prime to <code class="varname">n</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>
@@ -52,7 +52,7 @@
1 2 1]</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>
Return all factors of <code class="varname">n</code> in a vector. This
includes all the non-prime factors as well. It includes 1 and the
@@ -75,7 +75,7 @@
of two factors that are very close to each other.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Find the first primitive element in F<sub>q</sub>, the
finite
group of order <code class="varname">q</code>. Of course <code class="varname">q</code> must be a
prime.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Find a random primitive element in F<sub>q</sub>,
the finite
group of order <code class="varname">q</code> (q must be a prime).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of n in F<sub>q</sub>, the finite
@@ -99,7 +99,7 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -112,8 +112,8 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.
</p></dd><dt><span class="term"><a name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre
class="synopsis">IsOdd (n)</pre><p>Tests if an integer is odd.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Check an integer for being any perfect power, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
Check an integer for being a perfect square of an integer. The number must
- be a real integer. Negative integers are of course never perfect
- squares of real integers.
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
</p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>
Tests primality of integers, for numbers less than 2.5e10 the
answer is deterministic (if Riemann hypothesis is true). For
@@ -151,12 +151,12 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returns the <code class="varname">n</code>th Lucas number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Return all maximal prime power factors of a
number.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>
@@ -170,7 +170,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -185,7 +185,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
better on smaller integers.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
@@ -194,7 +194,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
result is deterministic.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Returns inverse of n mod m.</p><p>
diff --git a/help/pt_BR/html/ch11s08.html b/help/pt_BR/html/ch11s08.html
index 76a9802..29dc9a2 100644
--- a/help/pt_BR/html/ch11s08.html
+++ b/help/pt_BR/html/ch11s08.html
@@ -1,11 +1,11 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Manipulação de
matrizes</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Manual do Genius"><link rel="up" href="ch11.html" title="Capítulo 11. List of GEL
functions"><link rel="prev" href="ch11s07.html" title="Teoria dos números"><link rel="next"
href="ch11s09.html" title="Álgebra linear"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Manipulação de matrizes</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s07.html">Anterior</a> </td><th width="60%" align="center">Capítulo
11. List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s09.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><d
iv><div><h2 class="title" style="clear: both"><a name="genius-gel-function-list-matrix"></a>Manipulação de
matrizes</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Apply a function over all entries of a matrix and return a matrix of the
results.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Apply a function over all entries of 2 matrices (or 1
value and 1 matrix) and return a matrix of the results.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Gets
the columns of a matrix as a horizontal vector.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</
span></dt><dd><pre class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Remove column(s) and row(s) from a
matrix.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Calculate the kth compound matrix of A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
- Count the number of zero columns in a matrix. For example
- once your column reduce a matrix you can use this to find
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
</p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,col)</pre><p>Delete a column of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,row)</pre><p>Delete a row of a matrix.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Gets the diagonal entries of a matrix as a column vector.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
See
@@ -28,7 +28,7 @@
<strong class="userinput"><code>5</code></strong>, we return <strong
class="userinput"><code>[1,4,5]</code></strong>. If
<code class="varname">msize</code> is 0, we always return <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>É
uma matriz diagonal.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Check if a matrix is the identity matrix. Automatically returns <code
class="constant">false</code>
if the matrix is not square. Also works on numbers, in which
@@ -37,12 +37,12 @@
no error is generated and <code class="constant">false</code> is returned.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Is a matrix lower triangular. That is, are all the entries
above the diagonal zero.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Check if a matrix is non-negative, that is if each element
is non-negative.
Do not confuse positive matrices with positive semi-definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Check if a matrix is positive, that is if each element is
positive (and hence real). In particular, no element is 0. Do not confuse
positive matrices with positive definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Check if a matrix is a matrix of rational (non-complex)
numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Check if a matrix is a matrix of real (non-complex) numbers.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>
Check if a matrix is square, that is its width is equal to
@@ -62,7 +62,7 @@ functions make this check. Values can be any number including complex numbers.<
<strong class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> is the same as
<strong class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Make column vector out of matrix by putting columns above
each other. Returns <code class="constant">null</code> when given <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>
diff --git a/help/pt_BR/html/ch11s09.html b/help/pt_BR/html/ch11s09.html
index 109accb..639d18d 100644
--- a/help/pt_BR/html/ch11s09.html
+++ b/help/pt_BR/html/ch11s09.html
@@ -50,7 +50,7 @@ result as a vector and not added together.</p></dd><dt><span class="term"><a nam
diagonal).
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Get the eigenvectors of a square matrix. Optionally get also
@@ -58,7 +58,7 @@ the eigenvalues and their algebraic multiplicities.
Currently only works for matrices of size up to 2 by 2.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Apply the Gram-Schmidt process (to the columns) with respect to
@@ -152,7 +152,7 @@ determinant.
of two matrices.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
@@ -182,7 +182,7 @@ determinant.
and <code class="varname">U</code> to <code class="constant">null</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Get the <code class="varname">i</code>-<code class="varname">j</code>
minor of a matrix.</p><p>
@@ -218,7 +218,7 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<code class="varname">Q</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Return the Rayleigh quotient (also called the Rayleigh-Ritz
quotient or ratio) of a matrix and a vector.</p><p>
@@ -241,45 +241,45 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</p></dd><dt><span
class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Return the matrix corresponding
to rotation around origin in R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
x-axis.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (angle)</pre><p>Return the matrix corresponding to rotation around origin in
R<sup>3</sup> about the
y-axis.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
z-axis.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Get a basis matrix for the rowspace of a matrix.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Evaluate (v,w) with respect to the sesquilinear form given
by the matrix A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Return a function that evaluates two vectors with
respect to the sesquilinear form given by A.</p></dd><dt><span class="term"><a name="gel
-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returns the Smith normal form of a matrix over fields (will
end up with 1's on the diagonal).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Solve linear system Mx=V, return solution V if there
is a unique solution, <code class="constant">null</code> otherwise. Extra two reference parameters can
optionally be used to get the reduced M and V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Return the Toeplitz matrix constructed given the first column c
and (optionally) the first row r. If only the column c is given then it is
conjugated and the nonconjugated version is used for the first row to give a
Hermitian matrix (if the first element is real of course).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Aliases: <code class="function">trace</code></p><p>Calculate the trace of
a matrix. That is the sum of the diagonal elements.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Transpose of a matrix. This is the same as the
<strong class="userinput"><code>.'</code></strong> operator.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Aliases: <code class="function">vander</code></p><p>Return the
Vandermonde matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>The angle of two vectors with respect to inner product given by
<code class="varname">B</code>. If <code class="varname">B</code> is not given then the standard
Hermitian product is used. <code class="varname">B</code> can either be a sesquilinear
function of two arguments or it can be a matrix giving a sesquilinear form.
</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>The direct sum of the vector spaces M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersection of the subspaces given by M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>The sum of the vector spaces M and N, that is {w | w=m+n, m
in M, n in N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Aliases: <code class="function">Adjugate</code></p><p>Get the classical
adjoint (adjugate) of a matrix.</p></dd><dt><span class="term"><a name="gel-function-cref"></a>cref</spa
n></dt><dd><pre class="synopsis">cref (M)</pre><p>Aliases: <code class="function">CREF</code> <code
class="function">ColumnReducedEchelonForm</code></p><p>Compute the Column Reduced Echelon
Form.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Aliases: <code class="function">Determinant</code></p><p>Get the determinant
of a matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Aliases: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Get the row echelon form of a matrix. That is, apply gaussian
elimination but not backaddition to <code class="varname">M</code>. The pivot rows are
divided to make all pivots 1.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Aliases: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Get the reduced row echelon form of a matrix. That is,
apply gaussian elimination together with backaddition to <code class="varname">M</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s08.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Acima</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s10.html">Próxima</a></td></tr><tr><td width="40%" align="left"
valign="top">Manipulação de matrizes </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Principal</a></td><td width="40%" align="right" valign="top">
Combinatória</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s10.html b/help/pt_BR/html/ch11s10.html
index 1ecad7b..303e4e3 100644
--- a/help/pt_BR/html/ch11s10.html
+++ b/help/pt_BR/html/ch11s10.html
@@ -3,7 +3,10 @@
<a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Get all combinations of k numbers from 1 to n as a vector of vectors.
(See also <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)
-</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Double factorial: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
See
<a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Factorial: <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
@@ -20,17 +23,18 @@
<strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
- Calculate the Frobenius number. That is calculate smallest
+ Calculate the Frobenius number. That is calculate largest
number that cannot be given as a non-negative integer linear
combination of a given vector of non-negative integers.
The vector can be given as separate numbers or a single vector.
All the numbers given should have GCD of 1.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(combining_rule)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>
Find the vector <code class="varname">c</code> of non-negative integers
@@ -40,8 +44,18 @@
of non-negative integers.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coeffi
cients. Takes a vector of
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coefficients. Takes a vector of
<code class="varname">k</code>
non-negative integers and computes the multinomial coefficient.
This corresponds to the coefficient in the homogeneous polynomial
@@ -57,7 +71,7 @@
<strong class="userinput"><code>Binomial(a+b,b)</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Get combination that would come after v in call to
@@ -77,6 +91,9 @@ do (
) while not IsNull(n:=NextCombination(n,6));</code></strong>
</pre><p>
See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Get the Pascal's triangle as a matrix. This will return
an <code class="varname">i</code>+1 by <code class="varname">i</code>+1 lower diagonal
matrix that is the Pascal's triangle after <code class="varname">i</code>
@@ -86,7 +103,7 @@ do (
</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Get all permutations of <code class="varname">k</code> numbers from 1 to <code
class="varname">n</code> as a vector of vectors.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Aliases: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k =
n(n+1)...(n+(k-1)).</p><p>
See
<a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
@@ -109,5 +126,5 @@ do (
<code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Acima</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Álgebra
linear </td><td width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td
width="40%" align="right" valign="top"> Cálculo</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s11.html b/help/pt_BR/html/ch11s11.html
index 230b77d..721dfb5 100644
--- a/help/pt_BR/html/ch11s11.html
+++ b/help/pt_BR/html/ch11s11.html
@@ -25,7 +25,7 @@ the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, whil
<strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Either <code class="varname">a</code>
or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Try to calculate an infinite product for a single parameter
function.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Try to calculate an infinite product for a
double parameter function with func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Try to calculate an infinite sum for a single parameter function.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,inc)</pre><p>Try to calculate an infinite sum for a double
parameter function with func(arg,n).</p></dd><d
t><span class="term"><a name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre
class="synopsis">IsContinuous (f,x0)</pre><p>Try and see if a real-valued function is continuous at x0 by
calculating the limit there.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Test for differentiability by approximating the left and
right limits and comparing.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calculate the left limit of a real-valued function at x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Calculate the
limit of a real-valued function at x0. Tries to calculate both left and right limits.</p></dd><dt><span
class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</sp
an></dt><dd><pre class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integration by midpoint
rule.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Aliases: <code
class="function">NDerivative</code></p><p>Attempt to calculate numerical derivative.</p><p>
See
@@ -40,7 +40,7 @@ up to <code class="varname">N</code>th harmonic computed numerically. The coeff
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
@@ -50,7 +50,7 @@ trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the cosine Fourier series of
@@ -65,7 +65,7 @@ Note that <strong class="userinput"><code>a@(1)</code></strong> is
the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -76,7 +76,7 @@ only has cosine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the sine Fourier series of
@@ -88,7 +88,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -99,7 +99,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration by rule set in NumericalIntegralFunction of f
from a to b using NumericalIntegralSteps steps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Attempt to calculate numerical left
derivative.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Attempt to
calculate the limit of f(step_fun(i)) as i goes from 1 to N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">Nume
ricalRightDerivative (f,x0)</pre><p>Attempt to calculate numerical right derivative.</p></dd><dt><span
class="term"><a name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
<code class="function">f</code> with half period <code class="varname">L</code>. That
diff --git a/help/pt_BR/html/ch11s12.html b/help/pt_BR/html/ch11s12.html
index e4d704c..dcf5839 100644
--- a/help/pt_BR/html/ch11s12.html
+++ b/help/pt_BR/html/ch11s12.html
@@ -1,21 +1,21 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Funções</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Manual
do Genius"><link rel="up" href="ch11.html" title="Capítulo 11. List of GEL functions"><link rel="prev"
href="ch11s11.html" title="Cálculo"><link rel="next" href="ch11s13.html" title="Solução de
equações"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Funções</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s13.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="cl
ear: both"><a name="genius-gel-function-list-functions"></a>Funções</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Aliases:
<code class="function">Arg</code> <code class="function">arg</code></p><p>argument (angle) of complex
number.</p></dd><dt><span class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre
class="synopsis">BesselJ0 (x)</pre><p>Bessel function of the first kind of order 0. Only implemented for
real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Bessel
function of the first kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Bessel
function of the first kind of order <code class="varname">n</code>. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Bessel
function of the second kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Bessel
function of the second kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Bessel
function of the second kind of order <code class="varname">n</code>. Only implemented for real
numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Aliases: <code class="function">erf</code></p><p>The error function, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
@@ -27,7 +27,7 @@
</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Aliases: <code class="function">Gamma</code></p><p>The Gamma function. Currently only
implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returns 1 if and only if all elements are equal.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>
The principal branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>.
@@ -38,7 +38,7 @@
See <a class="link" href="ch11s12.html#gel-function-LambertWm1"><code
class="function">LambertWm1</code></a> for the other real branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>
The minus-one branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>
@@ -48,29 +48,34 @@
See <a class="link" href="ch11s12.html#gel-function-LambertW"><code
class="function">LambertW</code></a> for the principal branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Find the first value where f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Moebius mapping of the disk to itself mapping a to 0.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity
respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Moebius mapping using the cross ratio taking
infinity to infinity and z2,z3 to 1 and 0 respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 1 and z3,z4 to 0 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 0 and z2,z4 to 1 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poisson kernel on D(0,R) (not normalized to
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Aliases: <code class="function">zeta</code></p><p>The Riemann zeta
function. Currently only implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>The unit step function is 0 for x<0, 1 otherwise. This is the
integral of the Dirac Delta function. Also called the Heaviside function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>
The <code class="function">cis</code> function, that is the same as
<strong class="userinput"><code>cos(x)+1i*sin(x)</code></strong>
@@ -78,5 +83,5 @@
<strong class="userinput"><code>sin(x)/x</code></strong>.
If you want the normalized function call <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
</p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Acima</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Cálculo </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td width="40%" align="right"
valign="top"> Solução de equações</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s13.html b/help/pt_BR/html/ch11s13.html
index 7fb41e5..fdd2d42 100644
--- a/help/pt_BR/html/ch11s13.html
+++ b/help/pt_BR/html/ch11s13.html
@@ -10,7 +10,7 @@
See
<a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
@@ -29,12 +29,12 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
@@ -73,7 +73,7 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Find root of a function using the bisection method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
@@ -102,7 +102,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Find zeros using Newton's method. <code class="varname">f</code> is
the function and <code class="varname">df</code> is the derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
@@ -116,7 +116,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>
Compute roots of a polynomial (degrees 1 through 4)
using one of the formulas for such polynomials.
@@ -139,8 +139,9 @@
Returns a column vector of the two solutions.
</p><p>
See
- <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>
Compute roots of a quartic (degree 4) polynomial using the
quartic formula. The polynomial should be given as a
@@ -152,7 +153,7 @@
See
<a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
@@ -168,14 +169,14 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
@@ -209,5 +210,5 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Acima</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Funções </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td width="40%" align="right"
valign="top"> Estatística</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s14.html b/help/pt_BR/html/ch11s14.html
index c8aec9a..4822149 100644
--- a/help/pt_BR/html/ch11s14.html
+++ b/help/pt_BR/html/ch11s14.html
@@ -1,20 +1,27 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Estatística</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual do Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. List of GEL functions"><link rel="prev" href="ch11s13.html" title="Solução de
equações"><link rel="next" href="ch11s15.html" title="Polinômios"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Estatística</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-function-list-statistics"></a>Estatística</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average of an entire matrix.</p><p>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Estatística</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual do Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. List of GEL functions"><link rel="prev" href="ch11s13.html" title="Solução de
equações"><link rel="next" href="ch11s15.html" title="Polinômios"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Estatística</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Anterior</a> </td><th width="60%" align="center">Capítulo 11. List of GEL
functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-function-list-statistics"></a>Estatística</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Aliases:
<code class="function">average</code> <code class="function">Mean</code> <code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral of the GaussFunction from 0 to <code
class="varname">x</code> (area under the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>The normalized Gauss distribution function (the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Aliases: <code class="function">median</code></p><p>Calculate median of
an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix.</p><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calculate median of each row in a matrix and return a column
vector of the medians.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdevp</code></p><p>Calculate the population standard deviations of rows of a matrix and
return a vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdev</code></p><p>Calculate the standard deviations of rows of a matrix and return a
vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Aliases: <code class="function">stdev</code></p><p>Calculate
the standard deviation of a whole matrix.</p></dd></dl></div></div><div class="navfooter"><hr><table widt
h="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Acima</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Solução de equações
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td width="40%"
align="right" valign="top"> Polinômios</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s15.html b/help/pt_BR/html/ch11s15.html
index 24fd4fe..9caa20b 100644
--- a/help/pt_BR/html/ch11s15.html
+++ b/help/pt_BR/html/ch11s15.html
@@ -17,5 +17,5 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Take second polynomial (as vector)
derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Take polynomial (as vector) derivative.</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Make function out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Make string out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly
(p1,p2)</pre><p>Subtract two polynomials (as vectors).
</p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Trim zeros from a polynomial (as vector).</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s14.html">Anterior</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Acima</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Estatística </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td width="40%" align="right"
valign="top"> Teoria dos conjuntos</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s18.html b/help/pt_BR/html/ch11s18.html
index 03c6e95..fe79790 100644
--- a/help/pt_BR/html/ch11s18.html
+++ b/help/pt_BR/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscelânea</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual do Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. List of GEL functions"><link rel="prev" href="ch11s17.html" title="Álgebra
comutativa"><link rel="next" href="ch11s19.html" title="Operações simbólicas"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Miscelânea</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><th width="60%" align="center">Capítulo
11. List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="t
itle" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Miscelânea</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a vector of ASCII values.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a
string to a vector of 0-based alphabet values (positions in the alphabet string), -1's for unknown
letters.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Acima</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Álgebra
comutativa </td><td width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td
width="40%" align="right" valign="top"> Operações simbólicas</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Miscelânea</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Manual do Genius"><link rel="up" href="ch11.html"
title="Capítulo 11. List of GEL functions"><link rel="prev" href="ch11s17.html" title="Álgebra
comutativa"><link rel="next" href="ch11s19.html" title="Operações simbólicas"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Miscelânea</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><th width="60%" align="center">Capítulo
11. List of GEL functions</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Próxima</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="t
itle" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Miscelânea</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Anterior</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Acima</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Próxima</a></td></tr><tr><td width="40%" align="left" valign="top">Álgebra
comutativa </td><td width="20%" align="center"><a accesskey="h" href="index.html">Principal</a></td><td
width="40%" align="right" valign="top"> Operações simbólicas</td></tr></table></div></body></html>
diff --git a/help/pt_BR/html/ch11s20.html b/help/pt_BR/html/ch11s20.html
index 2b5ea88..f6b4ad0 100644
--- a/help/pt_BR/html/ch11s20.html
+++ b/help/pt_BR/html/ch11s20.html
@@ -102,7 +102,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
</pre><p>
@@ -153,7 +153,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
</pre><p>
@@ -330,7 +330,7 @@ limits as <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.
<code class="varname">n</code> by 3 matrix for a longer polyline.
</p><p>
Extra parameters can be added to specify line color, thickness,
- arrows, the plotting window, or legend.
+ the plotting window, or legend.
You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
<strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>,
diff --git a/help/pt_BR/html/index.html b/help/pt_BR/html/index.html
index c969d84..9a89635 100644
--- a/help/pt_BR/html/index.html
+++ b/help/pt_BR/html/index.html
@@ -1,5 +1,5 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Manual do
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Manual da ferramenta matemática Genius"><link rel="home" href="index.html" title="Manual do
Genius"><link rel="next" href="ch01.html" title="Capítulo 1. Introdução"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Manual do Genius</th></tr><tr><td width="20%"
align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a accesskey="n"
href="ch01.html">Próxima</a></td></tr></table><hr></div><div lang="pt_BR" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Manual do Genius</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><span cl
ass="firstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Universidade do Estado de Oklahoma<br></span><div class="address"><p> <code
class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code>
</p></div></div></div><div class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">Universidade de
Queensland, Austrália<br></span><div class="address"><p> <code class="email"><<a class="email"
href="mailto:kaiw itee uq edu au">kaiw itee uq edu au</a>></code>
</p></div></div></div></div></div><div><p class="releaseinfo">This manual describes version 1.0.22 of Genius.
- </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2013. Enrico
Nicoletto (liverig gmail com)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Permissão
concedida para copiar, distribuir e/ou modificar este documento sob os termos da Licença de Documentação
Livre GNU (GNU Free Documentation License), Versão 1.1 ou qualquer versão mais recente publicada pela Free
Software Foundation; sem Seções Invariantes, Textos de Capa Frontal, e sem Textos de Contracapa. Você pode
encontrar uma cópia da licença GFDL neste <a class="ulink" href="ghelp:fdl" target="_top">link</a> ou no
arquivo COPYING-DOCS distribuído com este manual.</p><p>Este manual é parte da coleção de manuais do GNOME
distribuídos sob a GFDL. Se você quiser distribuí-lo separadamente da coleção, você pode fazê-lo adicionando
ao manu
al uma cópia da licença, como descrito na seção 6 da licença.</p><p>Muitos dos nomes usados por empresas
para distinguir seus produtos e serviços são reivindicados como marcas registradas. Onde esses nomes aparecem
em qualquer documentação do GNOME e os membros do Projeto de Documentação do GNOME estiverem cientes dessas
marcas registradas, os nomes aparecerão impressos em letras maiúsculas ou com iniciais em maiúsculas.</p><p>O
DOCUMENTO E VERSÕES MODIFICADAS DO DOCUMENTO SÃO FORNECIDOS SOB OS TERMOS DA GNU FREE DOCUMENTATION LICENSE
COM O ENTENDIMENTO ADICIONAL DE QUE: </p><div class="orderedlist"><ol class="orderedlist" type="1"><li
class="listitem"><p>O DOCUMENTO É FORNECIDO NA BASE "COMO ESTÁ", SEM GARANTIAS DE QUALQUER TIPO, TANTO
EXPRESSA OU IMPLÍCITA, INCLUINDO, MAS NÃO LIMITADO A, GARANTIAS DE QUE O DOCUMENTO OU VERSÃO MODIFICADA DO
DOCUMENTO SEJA COMERCIALIZÁVEL, LIVRE DE DEFEITOS, PRÓPRIO PARA UM PROPÓSITO ESPECÍFICO OU SEM INFRAÇÕES. TO
DO O RISCO A RESPEITO DA QUALIDADE, EXATIDÃO, E DESEMPENHO DO DOCUMENTO OU VERSÕES MODIFICADAS DO DOCUMENTO
É DE SUA RESPONSABILIDADE. SE ALGUM DOCUMENTO OU VERSÃO MODIFICADA SE PROVAR DEFEITUOSO EM QUALQUER ASPECTO,
VOCÊ (NÃO O ESCRITOR INICIAL, AUTOR OU QUALQUER CONTRIBUIDOR) ASSUME O CUSTO DE QUALQUER SERVIÇO NECESSÁRIO,
REPARO OU CORREÇÃO. ESSA RENÚNCIA DE GARANTIAS CONSTITUI UMA PARTE ESSENCIAL DESTA LICENÇA. NENHUM USO DESTE
DOCUMENTO OU VERSÃO MODIFICADA DESTE DOCUMENTO É AUTORIZADO SE NÃO FOR SOB ESSA RENÚNCIA; E</p></li><li
class="listitem"><p>SOB NENHUMA CIRCUNSTÂNCIA E SOB NENHUMA TEORIA LEGAL, TANTO EM DANO (INCLUINDO
NEGLIGÊNCIA), CONTRATO, OU OUTROS, DEVEM O AUTOR, ESCRITOR INICIAL, QUALQUER CONTRIBUIDOR, OU QUALQUER
DISTRIBUIDOR DO DOCUMENTO OU VERSÃO MODIFICADA DO DOCUMENTO, OU QUALQUER FORNECEDOR DE ALGUMA DESSAS PARTES,
SEREM CONSIDERADOS RESPONSÁVEIS A QUALQUER PESSOA POR QUALQUER DANO, SEJA DIRETO, INDIRETO, ESPECIAL,
ACIDENTAL OU DANO
S DECORRENTES DE QUALQUER NATUREZA, INCLUINDO, MAS NÃO LIMITADO A, DANOS POR PERDA DE BOA VONTADE, TRABALHO
PARADO, FALHA OU MAU FUNCIONAMENTO DO COMPUTADOR, OU QUALQUER E TODOS OS OUTROS DANOS OU PERDAS RESULTANTES
OU RELACIONADOS AO USO DO DOCUMENTO E VERSÕES MODIFICADAS, MESMO QUE TAL PARTE TENHA SIDO INFORMADA DA
POSSIBILIDADE DE TAIS DANOS.</p></li></ol></div></div></div><div><div class="legalnotice"><a
name="idm45933042884336"></a><p class="legalnotice-title"><b>Comentários</b></p><p>
+ </p></div><div><p class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p
class="copyright">Copyright © 2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2013. Enrico
Nicoletto (liverig gmail com)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Permissão
concedida para copiar, distribuir e/ou modificar este documento sob os termos da Licença de Documentação
Livre GNU (GNU Free Documentation License), Versão 1.1 ou qualquer versão mais recente publicada pela Free
Software Foundation; sem Seções Invariantes, Textos de Capa Frontal, e sem Textos de Contracapa. Você pode
encontrar uma cópia da licença GFDL neste <a class="ulink" href="ghelp:fdl" target="_top">link</a> ou no
arquivo COPYING-DOCS distribuído com este manual.</p><p>Este manual é parte da coleção de manuais do GNOME
distribuídos sob a GFDL. Se você quiser distribuí-lo separadamente da coleção, você pode fazê-lo adicionando
ao manu
al uma cópia da licença, como descrito na seção 6 da licença.</p><p>Muitos dos nomes usados por empresas
para distinguir seus produtos e serviços são reivindicados como marcas registradas. Onde esses nomes aparecem
em qualquer documentação do GNOME e os membros do Projeto de Documentação do GNOME estiverem cientes dessas
marcas registradas, os nomes aparecerão impressos em letras maiúsculas ou com iniciais em maiúsculas.</p><p>O
DOCUMENTO E VERSÕES MODIFICADAS DO DOCUMENTO SÃO FORNECIDOS SOB OS TERMOS DA GNU FREE DOCUMENTATION LICENSE
COM O ENTENDIMENTO ADICIONAL DE QUE: </p><div class="orderedlist"><ol class="orderedlist" type="1"><li
class="listitem"><p>O DOCUMENTO É FORNECIDO NA BASE "COMO ESTÁ", SEM GARANTIAS DE QUALQUER TIPO, TANTO
EXPRESSA OU IMPLÍCITA, INCLUINDO, MAS NÃO LIMITADO A, GARANTIAS DE QUE O DOCUMENTO OU VERSÃO MODIFICADA DO
DOCUMENTO SEJA COMERCIALIZÁVEL, LIVRE DE DEFEITOS, PRÓPRIO PARA UM PROPÓSITO ESPECÍFICO OU SEM INFRAÇÕES. TO
DO O RISCO A RESPEITO DA QUALIDADE, EXATIDÃO, E DESEMPENHO DO DOCUMENTO OU VERSÕES MODIFICADAS DO DOCUMENTO
É DE SUA RESPONSABILIDADE. SE ALGUM DOCUMENTO OU VERSÃO MODIFICADA SE PROVAR DEFEITUOSO EM QUALQUER ASPECTO,
VOCÊ (NÃO O ESCRITOR INICIAL, AUTOR OU QUALQUER CONTRIBUIDOR) ASSUME O CUSTO DE QUALQUER SERVIÇO NECESSÁRIO,
REPARO OU CORREÇÃO. ESSA RENÚNCIA DE GARANTIAS CONSTITUI UMA PARTE ESSENCIAL DESTA LICENÇA. NENHUM USO DESTE
DOCUMENTO OU VERSÃO MODIFICADA DESTE DOCUMENTO É AUTORIZADO SE NÃO FOR SOB ESSA RENÚNCIA; E</p></li><li
class="listitem"><p>SOB NENHUMA CIRCUNSTÂNCIA E SOB NENHUMA TEORIA LEGAL, TANTO EM DANO (INCLUINDO
NEGLIGÊNCIA), CONTRATO, OU OUTROS, DEVEM O AUTOR, ESCRITOR INICIAL, QUALQUER CONTRIBUIDOR, OU QUALQUER
DISTRIBUIDOR DO DOCUMENTO OU VERSÃO MODIFICADA DO DOCUMENTO, OU QUALQUER FORNECEDOR DE ALGUMA DESSAS PARTES,
SEREM CONSIDERADOS RESPONSÁVEIS A QUALQUER PESSOA POR QUALQUER DANO, SEJA DIRETO, INDIRETO, ESPECIAL,
ACIDENTAL OU DANO
S DECORRENTES DE QUALQUER NATUREZA, INCLUINDO, MAS NÃO LIMITADO A, DANOS POR PERDA DE BOA VONTADE, TRABALHO
PARADO, FALHA OU MAU FUNCIONAMENTO DO COMPUTADOR, OU QUALQUER E TODOS OS OUTROS DANOS OU PERDAS RESULTANTES
OU RELACIONADOS AO USO DO DOCUMENTO E VERSÕES MODIFICADAS, MESMO QUE TAL PARTE TENHA SIDO INFORMADA DA
POSSIBILIDADE DE TAIS DANOS.</p></li></ol></div></div></div><div><div class="legalnotice"><a
name="idm51"></a><p class="legalnotice-title"><b>Comentários</b></p><p>
To report a bug or make a suggestion regarding the <span class="application">Genius Mathematics
Tool</span>
application or this manual, please visit the
<a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius
diff --git a/help/ru/html/ch05s07.html b/help/ru/html/ch05s07.html
index 6f56e9a..f1fac39 100644
--- a/help/ru/html/ch05s07.html
+++ b/help/ru/html/ch05s07.html
@@ -32,10 +32,12 @@ returns 3.
numbers, but operates element by element on matrices.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a\b</code></strong></span></dt><dd><p>Обратное деление. Это то же самое, что <strong
class="userinput"><code>b/a</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Поэлементное обратное
деление.</p></dd><dt><span class="term"><strong class="userinput"><code>a%b</code></strong></span></dt><dd><p>
The mod operator. This does not turn on the <a class="link" href="ch05s06.html" title="Modular
Evaluation">modular mode</a>, but
- just returns the remainder of <strong class="userinput"><code>a/b</code></strong>.
+ just returns the remainder of integer division
+ <strong class="userinput"><code>a/b</code></strong>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>
- Element by element the mod operator. Returns the remainder
- after element by element integer <strong class="userinput"><code>a./b</code></strong>.
+ Element by element mod operator. Returns the remainder
+ after element by element integer division
+ <strong class="userinput"><code>a./b</code></strong>.
</p></dd><dt><span class="term"><strong class="userinput"><code>a mod
b</code></strong></span></dt><dd><p>
Modular evaluation operator. The expression <code class="varname">a</code>
is evaluated modulo <code class="varname">b</code>. See <a class="xref" href="ch05s06.html"
title="Modular Evaluation">«Modular Evaluation»</a>.
@@ -57,21 +59,21 @@ returns 3.
greater than or equal to
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a >= b >= c</code></strong>
- (can also be combine with the greater than operator).
+ (and they can also be combined with the greater than operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>
Less than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
less than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a < b < c</code></strong>
- (can also be combine with the less than or equal to operator).
+ (they can also be combined with the less than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>
Greater than operator,
returns <code class="constant">true</code> if <code class="varname">a</code> is
greater than
<code class="varname">b</code> else returns <code class="constant">false</code>.
These can be chained as in <strong class="userinput"><code>a > b > c</code></strong>
- (can also be combine with the greater than or equal to operator).
+ (they can also be combined with the greater than or equal to operator).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span></dt><dd><p>Оператор сравнения. Если <code
class="varname">a</code> равно <code class="varname">b</code>, возвращает 0; если <code
class="varname">a</code> меньше <code class="varname">b</code>, возвращает -1; если <code
class="varname">a</code> больше <code class="varname">b</code>, возвращает 1.</p></dd><dt><span
class="term"><strong class="userinput"><code>a and b</code></strong></span></dt><dd><p>
Logical and. Returns true if both
<code class="varname">a</code> and <code class="varname">b</code> are true,
@@ -85,12 +87,12 @@ returns 3.
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>
Logical xor.
- Returns true exactly one of
+ Returns true if exactly one of
<code class="varname">a</code> or <code class="varname">b</code> is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>
- Logical not. Returns the logical negation of <code class="varname">a</code>
+ Logical not. Returns the logical negation of <code class="varname">a</code>.
</p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>
Negation operator. Returns the negative of a number or a matrix (works element-wise on a
matrix).
</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>
@@ -109,7 +111,7 @@ returns 3.
Get element of a matrix in row <code class="varname">b</code> and column
<code class="varname">c</code>. If <code class="varname">b</code>,
<code class="varname">c</code> are vectors, then this gets the corresponding
- rows columns or submatrices.
+ rows, columns or submatrices.
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>
Get row of a matrix (or multiple rows if <code class="varname">b</code> is a vector).
</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>То же, что и выше.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Возвращает столбец
матрицы (или столбцы, если <code class="varname">c</code> является вектором).</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>То же, что и
выше.</p></dd><dt><span class="term"><strong class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>
@@ -148,8 +150,13 @@ returns 3.
<strong class="userinput"><code>float(1:2/5:3)</code></strong> even gives you floating
point numbers and is ever so slightly more precise than
<strong class="userinput"><code>1.0:0.4:3.0</code></strong>.
- </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Создаёт мнимое число (умножает <code
class="varname">a</code> на мнимую единицу). Обратите внимание, что обчыно мнимая единица <code
class="varname">i</code> записывается в виде <strong class="userinput"><code>1i</code></strong>. Поэтому
вышеуказанное выражение эквивалентно </p><pre class="programlisting">(a)*1i
- </pre></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>
+ Make <code class="varname">a</code> into an imaginary number (multiply <code
class="varname">a</code> by the
+ imaginary). Normally the imaginary number <code class="varname">i</code> is
+ written as <strong class="userinput"><code>1i</code></strong>. So the above is equal to
+ </p><pre class="programlisting">(a)*1i
+ </pre><p>
+ </p></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>
Quote an identifier so that it doesn't get evaluated. Or
quote a matrix so that it doesn't get expanded.
</p></dd><dt><span class="term"><strong class="userinput"><code>a swapwith
b</code></strong></span></dt><dd><p>Меняет местами значение <code class="varname">a</code> со значением <code
class="varname">b</code>. В настоящее время не работает с диапазонами элементов матрицы. Возвращает <code
class="constant">null</code>. Доступен, начиная с версии 1.0.13.</p></dd><dt><span class="term"><strong
class="userinput"><code>increment a</code></strong></span></dt><dd><p>Инкремент переменной <code
class="varname">a</code> на 1. Если <code class="varname">a</code> — матрица, то инкрементирует каждый
элемент. Это эквивалентно <strong class="userinput"><code>a=a+1</code></strong>, но немного быстрее.
Возвращает <code class="constant">null</code>. Доступ
ен с версии 1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment a by
b</code></strong></span></dt><dd><p>Инкремент переменной <code class="varname">a</code> на величину <code
class="varname">b</code>. Если <code class="varname">a</code> — матрица, то инкрементирует каждый элемент.
Это эквивалентно <strong class="userinput"><code>a=a+b</code></strong>, но немного быстрее. Возвращает <code
class="constant">null</code>. Доступен с версии 1.0.13.</p></dd></dl></div><div class="note"
style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Примечание</h3><p>Оператор @() делает
использование оператора : наиболее полезным. С его помощью можно указывать области матрицы. Таким образом,
a@(2:4,6) — �
�то строки 2,3,4 столбца 6. Или a@(,1:2) возвращает два первых столбца матрицы. Можно также присваивать
значения оператору @(), если правое значение — это матрица, совпадающая по размеру с данной областью, или
если это любой другой тип значений.</p></div><div class="note" style="margin-left: 0.5in; margin-right:
0.5in;"><h3 class="title">Примечание</h3><p>
diff --git a/help/ru/html/ch06s05.html b/help/ru/html/ch06s05.html
index ffe660b..496dc94 100644
--- a/help/ru/html/ch06s05.html
+++ b/help/ru/html/ch06s05.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Глобальные переменные
и область видимости переменных</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Руководство пользователя Genius"><link rel="up" href="ch06.html"
title="Глава 6. Программирование в GEL"><link rel="prev" href="ch06s04.html" title="Операторы
сравнения"><link rel="next" href="ch06s06.html" title="Parameter variables"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Глобальные переменные и область видимости
переменных</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch06s04.html">Пред.</a> </
td><th width="60%" align="center">Глава 6. Программирование в GEL</th><td width="20%" align="right"> <a
accesskey="n" href="ch06s06.html">След.</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Глобальные переменные и область видимости
переменных</h2></div></div></div><p>
GEL is a
- <a class="ulink" href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top">
dynamically scoped language</a>. We will explain what this
means below. That is, normal variables and functions are dynamically
scoped. The exception are
diff --git a/help/ru/html/ch07s02.html b/help/ru/html/ch07s02.html
index 932b312..b701823 100644
--- a/help/ru/html/ch07s02.html
+++ b/help/ru/html/ch07s02.html
@@ -3,10 +3,32 @@
the top level versus when they are inside parentheses or
inside functions. On the top level, enter acts the same as if
you press return on the command line. Therefore think of programs
- as just sequence of lines as if were entered on the command line.
+ as just a sequence of lines as if they were entered on the command line.
In particular, you do not need to enter the separator at the end of the
line (unless it is of course part of several statements inside
- parentheses).
+ parentheses). When a statement does not end with a separator on the
+ top level, the result is printed after being executed.
+ </p><p>
+ For example,
+ </p><pre class="programlisting">function f(x)=x^2
+f(3)
+</pre><p>
+ will print first the result of setting a function (a representation of
+ the function, in this case <code class="computeroutput">(`(x)=(x^2))</code>)
+ and then the expected 9. To avoid this, enter a separator
+ after the function definition.
+ </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p>
+ If you need to put a separator into your function then you have to surround with
+ parenthesis. For example:
+</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>
</p><p>
The following code will produce an error when entered on the top
level of a program, while it will work just fine in a function.
diff --git a/help/ru/html/ch11s04.html b/help/ru/html/ch11s04.html
index 0bb4db4..91f9b23 100644
--- a/help/ru/html/ch11s04.html
+++ b/help/ru/html/ch11s04.html
@@ -2,26 +2,26 @@
Catalan's Constant, approximately 0.915... It is defined to be the series where terms are
<strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, where <code class="varname">k</code>
ranges from 0 to infinity.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Catalan%27s_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/CatalansConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Aliases: <code class="function">gamma</code></p><p>
Euler's constant gamma. Sometimes called the
Euler-Mascheroni constant.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>The
Golden Ratio.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/GoldenRatio"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre
class="synopsis">Gravity</pre><p>Free fall acceleration at sea level in meters per second squared. This is
the standard gravity constant 9.80665. The gravity
in your particular neck of the woods might be different due to different altitude and the
fact that the earth is not perfectly
round and uniform.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Standard_gravity";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>
The base of the natural logarithm. <strong class="userinput"><code>e^x</code></strong>
is the exponential function
@@ -30,12 +30,12 @@
several numbers that are also called Euler's. An example is the gamma constant: <a class="link"
href="ch11s04.html#gel-function-EulerConstant"><code class="function">EulerConstant</code></a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/E"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/e.html"; target="_top">Mathworld</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-pi"></a>pi</span></dt><dd><pre
class="synopsis">pi</pre><p>Число «пи» — отношение длины окружности к её диаметру. Значение приблизительно
равно 3.14159265359...</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Pi"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> for more
information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s03.html">Пред.</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s05.html">След.</a></td></tr><tr><td width="40%" align="left" valign="top">Параметры
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Начало</a></td><td width="40%"
align="right" valign="top"> Числовые</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s05.html b/help/ru/html/ch11s05.html
index 90a4e32..ce7a220 100644
--- a/help/ru/html/ch11s05.html
+++ b/help/ru/html/ch11s05.html
@@ -5,7 +5,7 @@
to <strong class="userinput"><code>|x|</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolute
value)</a>,
<a class="ulink" href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath
(modulus)</a>,
<a class="ulink" href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld
(absolute value)</a> or
@@ -14,16 +14,16 @@ for more information.
</p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Заменяет очень малое число нулём.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Aliases: <code class="function">conj</code> <code
class="function">Conj</code></p><p>Calculates the complex conjugate of the complex number <code
class="varname">z</code>. If <code class="varname">z</code> is a vector or matrix,
all its elements are conjugated.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></dt><dd><pre class="synopsis">Denominator
(x)</pre><p>Возвращает знаменатель рационального числа.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Возвращает дробную часть числа.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Синонимы: <code class="function">ImaginaryPart</code></p><p>Get the imaginary
part of a complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields
4.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre class="synopsis">IntegerQuotient
(m,n)</pre><p>Деление без остатка.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(num)</pre><p>Check if argument is a complex (non-real) number. Do note that we really mean nonreal number.
That is,
<strong class="userinput"><code>IsComplex(3)</code></strong> yields false, while
<strong class="userinput"><code>IsComplex(3-1i)</code></strong> yields true.</p></dd><dt><span
class="term"><a name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (num)</pre><p>Check if argument is a possibly complex rational number.
That is, if both real and imaginary parts are
@@ -31,10 +31,10 @@ all its elements are conjugated.</p><p>
the form <strong class="userinput"><code>n+1i*m</code></strong> where <code
class="varname">n</code> and <code class="varname">m</code>
are integers.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(num)</pre><p>Проверяет, является ли аргумент целым числом (не комплексным).</p></dd><dt><span
class="term"><a name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (num)</pre><p>Check if argument is a non-negative real integer. That
is, either a positive integer or zero.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (num)</pre><p>Синонимы: <code
class="function">IsNaturalNumber</code></p><p>Проверяет, является ли аргумент положительным действительным
целым числом. Обратите внимание, что мы придерживаемся с�
�глашения о том, что 0 не является натуральным числом.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(num)</pre><p>Проверяет, является ли аргумент рациональным числом (не комплексным). Разумеется,
«рациональное» означает просто «не хранящееся в виде числа с плавающей точкой».</p></dd><dt><span
class="term"><a name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal
(num)</pre><p>Проверяет, является ли аргумент действительным числом.</p></dd><dt><span class="term"><a
name="gel-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator
(x)</pre><p>Возвращает числитель рационального числа.</p><
p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Re"></a>Re</span></dt><dd><pre
class="synopsis">Re (z)</pre><p>Синонимы: <code class="function">RealPart</code></p><p>Get the real part of a
complex number. For example <strong class="userinput"><code>Re(3+4i)</code></strong> yields 3.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Синонимы: <code class="function">sign</code></p><p>Возвращает знак числа.
То есть, возвращает <code class="literal">-1</code>, если значение отрицательно, <code
class="literal">0</code>, если рано нулю и <code class="literal">1</code>, если значение положительно. Если
<code class="varname">x</code> — комплексное число, то <code class="function">Sign</code> возвращает
направление на числовой оси (положительное или отрицательное) или 0.</p></dd><dt><span class="term"><a
name="gel-function-ceil"></a>ceil</span></dt><dd><pre class="synopsis">ceil (x)</pre><p>Синонимы: <code
class="function">Ceiling</code></p><p>Возвращает наименьше
е целое число, которое больше или равно <code class="varname">n</code>. Примеры: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ceil(1.1)</code></strong>
= 2
<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1.1)</code></strong>
@@ -48,12 +48,12 @@ all its elements are conjugated.</p><p>
exact arithmetic.
</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>Экспоненциальная функция. Это функция <strong
class="userinput"><code>e^x</code></strong>, где <code class="varname">e</code> — <a class="link"
href="ch11s04.html#gel-function-e">основание натурального логарифма</a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Exponential_function";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-float"></a>float</span></dt><dd><pre
class="synopsis">float (x)</pre><p>Возвращает представление числа <code class="varname">x</code> в виде числа
с плавающей точкой.</p></dd><dt><span class="term"><a name="gel-function-floor"></a>floor</span></dt><dd><pre
class="synopsis">floor (x)</pre><p>Синонимы: <code class="function">Floor</code></p><p>Возвращает наибольшее
целое число, которое меньше или равно <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-ln"></a>ln</span></dt><dd><pre class="synopsis">ln (x)</pre><p>Натуральный логарифм
(логарифм по основанию <code class="varname">e</code>).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NaturalLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logarithm of <code
class="varname">x</code> base <code class="varname">b</code> (calls <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> if in modulo
mode), if base is not given, <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a> is used.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Логарифм <code
class="varname">x</code> по основанию 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis">log2 (x)</pre><p>Синоним: <code
class="function">lg</code></p><p>Логарифм <code class="varname">x</code> по основанию 2.</p></dd><dt><span
class="term"><a name="gel-f
unction-max"></a>max</span></dt><dd><pre class="synopsis">max (a,args...)</pre><p>Псевдонимы: <code
class="function">Max</code><code class="function">Maximum</code></p><p>Возвращает максимальный из аргументов
или элементов матрицы.</p></dd><dt><span class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre
class="synopsis">min (a,args...)</pre><p>Псевдонимы: <code class="function">Min</code><code
class="function">Minimum</code></p><p>Возвращает минимальный из аргументов или элементов
матрицы.</p></dd><dt><span class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre
class="synopsis">rand (size...)</pre><p>Генерирует случайное число с плавающей точкой в диапазоне <code
class="literal">[0,1)</code>. Если задан аргумент size, то может возвращать ма
трицу (если указано два числа) или вектор (если указано одно число) заданной размерности.</p></dd><dt><span
class="term"><a name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,size...)</pre><p>Генерирует случайное целое число в диапазоне <code class="literal">[0,max)</code>. Если
задан аргумент size, возвращает матрицу (если указано два числа) или вектор (если указано одно число)
заданной размерности. Например, </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
diff --git a/help/ru/html/ch11s06.html b/help/ru/html/ch11s06.html
index 2c24fdb..60d8a5c 100644
--- a/help/ru/html/ch11s06.html
+++ b/help/ru/html/ch11s06.html
@@ -1,6 +1,6 @@
<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Тригонометрические</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Руководство пользователя Genius"><link rel="up"
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title="Числовые"><link rel="next" href="ch11s07.html" title="Теория чисел"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Тригонометрические</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s05.html">Пред.</a> </td><th width="60%"
align="center">Глава 11. Список функций GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.ht
ml">След.</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
class="title" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Тригонометрические</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Синонимы: <code
class="function">arccos</code></p><p>Функция arccos (арккосинус, обратный косинус).</p></dd><dt><span
class="term"><a name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh
(x)</pre><p>Синонимы: <code class="function">arccosh</code></p><p>Функция arccosh (обратный гиперболический
косинус).</p></dd><dt><span class="term"><a name="gel-function-acot"></a>acot</span></dt><dd><pre
class="synopsis">acot (x)</pre><p>Синонимы: <code class="function"
arccot</code></p><p>Фунция arccot (арккотангенс, обратный котангенс).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Синонимы: <code
class="function">arccoth</code></p><p>Функция arccoth (обратный гиперболический
котангенс).</p></dd><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre
class="synopsis">acsc (x)</pre><p>Синонимы: <code class="function">arccsc</code></p><p>Обратный
косеканс.</p></dd><dt><span class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre
class="synopsis">acsch (x)</pre><p>Синонимы: <code class="function">arccsch</code></p><p>Обратный
гиперболический косеканс.</p></dd><dt><span class="term"><a
name="gel-function-asec"></a>asec</span></dt><dd><pre class="synopsis">asec (x)</pre><p>Синон
имы: <code class="function">arcsec</code></p><p>Обратный секанс.</p></dd><dt><span class="term"><a
name="gel-function-asech"></a>asech</span></dt><dd><pre class="synopsis">asech (x)</pre><p>Синонимы: <code
class="function">arcsech</code></p><p>Обратный гиперболический секанс.</p></dd><dt><span class="term"><a
name="gel-function-asin"></a>asin</span></dt><dd><pre class="synopsis">asin (x)</pre><p>Синонимы: <code
class="function">arcsin</code></p><p>Функция arcsin (арксинус, обратный синус).</p></dd><dt><span
class="term"><a name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh
(x)</pre><p>Синонимы: <code class="function">arcsinh</code></p><p>Фунция arcsinh (обратный гиперболический
синус).</p></dd><dt><span class="term"><a name="gel-function-atan"></a>atan</span></dt><dd><pre
class="synopsis">atan (x)</pre><p>Синони
мы: <code class="function">arctan</code></p><p>Вычисляет функцию arctan (арктангенс, обратный
тангенс).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Синонимы: <code class="function">arctanh</code></p><p>Функция arctanh
(обратный гиперболический тангенс).</p></dd><dt><span class="term"><a
name="gel-function-atan2"></a>atan2</span></dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Синонимы: <code
class="function">arctan2</code></p><p>Calculates the arctan2 function. If
<strong class="userinput"><code>x>0</code></strong> then it returns
@@ -11,11 +11,11 @@
rather than failing.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cos"></a>cos</span></dt><dd><pre
class="synopsis">cos (x)</pre><p>Вычисляет косинус.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-cosh"></a>cosh</span></dt><dd><pre
class="synopsis">cosh (x)</pre><p>Вычисляет гиперболический косинус.</p><p>
See
@@ -23,7 +23,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-cot"></a>cot</span></dt><dd><pre
class="synopsis">cot (x)</pre><p>Вычисляет котангенс.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Вычисляет гиперболический котангенс.</p><p>
See
@@ -31,7 +31,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-csc"></a>csc</span></dt><dd><pre
class="synopsis">csc (x)</pre><p>Вычисляет косеканс.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Вычисляет гиперболический косеканс.</p><p>
See
@@ -39,7 +39,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Вычисляет секанс.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Вычисляет гиперболический секанс.</p><p>
See
@@ -47,7 +47,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Вычисляет синус.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x)</pre><p>Вычисляет гиперболический синус.</p><p>
See
@@ -55,7 +55,7 @@
<a class="ulink" href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Вычисляет тангенс.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Вычисляет гиперболический тангенс.</p><p>
See
diff --git a/help/ru/html/ch11s07.html b/help/ru/html/ch11s07.html
index e6bb416..f0bea14 100644
--- a/help/ru/html/ch11s07.html
+++ b/help/ru/html/ch11s07.html
@@ -8,14 +8,14 @@
<a class="ulink" href="http://mathworld.wolfram.com/RelativelyPrime.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</pre><p>Return the <code class="varname">n</code>th Bernoulli number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bernoulli_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/BernoulliNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Aliases: <code class="function">CRT</code></p><p>Find the
<code class="varname">x</code> that solves the system given by
the vector <code class="varname">a</code> and modulo the elements of
<code class="varname">m</code>, using the Chinese Remainder Theorem.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ChineseRemainderTheorem";
target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/ChineseRemainderTheorem.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Given two factorizations, give the factorization of the
@@ -23,7 +23,7 @@
F<sub>q</sub>, the finite field of order <code class="varname">q</code>, where <code
class="varname">q</code>
is a prime, using the Silver-Pohlig-Hellman algorithm.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Проверяет делимость (делится ли <code class="varname">n</code> на
<code class="varname">m</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</pre><p>
@@ -32,7 +32,7 @@
relatively prime to <code class="varname">n</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/TotientFunction.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Возвращает <strong class="userinput"><code>n/d</code></strong>, но только если <code
class="varname">n</code> делится на <code class="varname">d</code>. Если не делится, то функция возвращает
мусор. Для очень больших чисел это гораздо быстрее, чем операция <strong
class="userinput"><code>n/d</code></strong>, но, разумеется, полезно только в том случае, если вы точно
знаете, что числа делятся без остатка.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize (n)</pre><p>
@@ -45,7 +45,7 @@
1 2 1]</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a name="gel-function-Factors"></a>Factors</span></dt><dd><pre
class="synopsis">Factors (n)</pre><p>
Return all factors of <code class="varname">n</code> in a vector. This
includes all the non-prime factors as well. It includes 1 and the
@@ -68,7 +68,7 @@
of two factors that are very close to each other.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Find the first primitive element in F<sub>q</sub>, the
finite
group of order <code class="varname">q</code>. Of course <code class="varname">q</code> must be a
prime.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Find a random primitive element in F<sub>q</sub>,
the finite
group of order <code class="varname">q</code> (q must be a prime).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Compute discrete log base <code class="varname">b</code> of n in F<sub>q</sub>, the finite
@@ -92,7 +92,7 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -105,8 +105,8 @@ precalculated and returned in the second column.</p></dd><dt><span class="term">
<a class="link" href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.
</p></dd><dt><span class="term"><a name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre
class="synopsis">IsOdd (n)</pre><p>Проверяет, является ли целое число нечётным.</p></dd><dt><span
class="term"><a name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre
class="synopsis">IsPerfectPower (n)</pre><p>Check an integer for being any perfect power,
a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span></dt><dd><pre class="synopsis">IsPerfectSquare
(n)</pre><p>
Check an integer for being a perfect square of an integer. The number must
- be a real integer. Negative integers are of course never perfect
- squares of real integers.
+ be an integer. Negative integers are of course never perfect
+ squares of integers.
</p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>
Tests primality of integers, for numbers less than 2.5e10 the
answer is deterministic (if Riemann hypothesis is true). For
@@ -144,12 +144,12 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasLhemer"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returns the <code class="varname">n</code>th Lucas number.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Return all maximal prime power factors of a
number.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>
@@ -163,7 +163,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
<a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/MersennePrime.html";
target="_top">Mathworld</a> or
<a class="ulink" href="http://www.mersenne.org/"; target="_top">GIMPS</a>
@@ -178,7 +178,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
better on smaller integers.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>
@@ -187,7 +187,7 @@ If <code class="varname">q</code> is not prime results are bogus.</p></dd><dt><s
result is deterministic.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a>,
or
<a class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre
class="synopsis">ModInvert (n,m)</pre><p>Returns inverse of n mod m.</p><p>
diff --git a/help/ru/html/ch11s08.html b/help/ru/html/ch11s08.html
index 3f11bb9..9f0746d 100644
--- a/help/ru/html/ch11s08.html
+++ b/help/ru/html/ch11s08.html
@@ -1,11 +1,11 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Операции с
матрицами</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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header"><tr><th colspan="3" align="center">Операции с матрицами</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s07.html">Пред.</a> </td><th width="60%" align="center">Глава 11. Список функций
GEL</th><td width="20%" align="right"> <a accesskey="
n" href="ch11s09.html">След.</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Операции с матрицами</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,func)</pre><p>Применяет функцию к каждому элементу матрицы и возвращает матрицу
результатов.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,func)</pre><p>Применяет функцию к каждому элементу двух матриц (или 1
значению и 1 матрице) и возвращает матрицу результатов.</p></dd><dt
<span class="term"><a name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre
class="synopsis">ColumnsOf (M)</pre><p>Возвращает столбцы матрицы в виде горизонтального
вектора.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>ComplementSubmatrix</span></dt><dd><pre
class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Remove column(s) and row(s) from a
matrix.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Calculate the kth compound matrix of A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>
- Count the number of zero columns in a matrix. For example
- once your column reduce a matrix you can use this to find
+ Count the number of zero columns in a matrix. For example,
+ once you column-reduce a matrix, you can use this to find
the nullity. See <a class="link" href="ch11s09.html#gel-function-cref"><code
class="function">cref</code></a>
and <a class="link" href="ch11s09.html#gel-function-Nullity"><code
class="function">Nullity</code></a>.
</p></dd><dt><span class="term"><a
name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre class="synopsis">DeleteColumn
(M,столбец)</pre><p>Удаляет столбец матрицы.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow
(M,строка)</pre><p>Удаляет строку матрицы.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Gets the diagonal entries of a matrix as a column vector.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Get the dot product of two vectors. The vectors must be of the
same size. No conjugates are taken so this is a bilinear form even if working over the
complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a> for the standard
sesquilinear inner product.</p><p>
See
@@ -28,7 +28,7 @@
<strong class="userinput"><code>5</code></strong>, we return <strong
class="userinput"><code>[1,4,5]</code></strong>. If
<code class="varname">msize</code> is 0, we always return <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal
(M)</pre><p>Является ли матрица диагональной.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt><dd><pre class="synopsis">IsIdentity
(x)</pre><p>Check if a matrix is the identity matrix. Automatically returns <code
class="constant">false</code>
if the matrix is not square. Also works on numbers, in which
@@ -37,15 +37,15 @@
no error is generated and <code class="constant">false</code> is returned.</p></dd><dt><span
class="term"><a name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Является ли матрица нижнетреугольной, то есть все её элементы
над диагональю равны нулю.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Check if a matrix is a matrix of integers (non-complex).</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Check if a matrix is non-negative, that is if each element
is non-negative.
Do not confuse positive matrices with positive semi-definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Check if a matrix is positive, that is if each element is
positive (and hence real). In particular, no element is 0. Do not confuse
positive matrices with positive definite matrices.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixRational"></a>IsMatrixRational</span></dt><dd><pre
class="synopsis">IsMatrixRational (M)</pre><p>Проверяет, является ли матрица матрицей из рациональных (не
комплексных) чисел.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre class="synopsis">IsMatrixReal
(M)</pre><p>Проверяет, является ли матрица матрицей из действительных (не комплексных)
чисел.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>Проверяет, является ли матрица квадратной, то есть её ширина равна высоте.</p></dd><dt><span
class="term"><a name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</sp
an></dt><dd><pre class="synopsis">IsUpperTriangular (M)</pre><p>Is a matrix upper triangular? That is, a
matrix is upper triangular if all the entries below the diagonal are zero.</p></dd><dt><span class="term"><a
name="gel-function-IsValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly
(M)</pre><p>Проверяет, состоит ли матрица только из чисел. Многие встроенные функции делают эту проверку.
Значения могут быть любыми числами, включая комплексные.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Является
ли аргумент горизонтальным или вертикальным вектором. Genius не делает различий между матрицей и вектором:
вектор — это просто матр
ица 1 на <code class="varname">n</code> или <code class="varname">n</code> на 1.</p></dd><dt><span
class="term"><a name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero
(x)</pre><p>Проверяет, состоит ли матрица из одних нулей. Также работает с числами, в этом случае
эквивалентна выражению <strong class="userinput"><code>x==0</code></strong>. Если переменная <code
class="varname">x</code> равна <code class="constant">null</code> (можно представить это, как матрицу 0 на 0
элементов), ошибка не генерируется и возвращается <code class="constant">true</code>, так как условие
является бессмысленным.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTr
iangular (M)</pre><p>Возвращает копию матрицы <code class="varname">M</code>, в которой все элементы под
диагональю заменены нулями.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Псевдоним: <code class="function">diag</code></p><p>Создаёт диагональную матрицу из
вектора. Значения для диагонали также могут быть переданы в виде аргументов функции. Поэтому <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> — то же самое, что и <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Make column vector out of matrix by putting columns above
each other. Returns <code class="constant">null</code> when given <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre class="synopsis">MatrixProduct
(A)</pre><p>Вычисляет произведение всех элементов матрицы или вектора. То есть, умножает друг на друга все
элементы и возвращает число, являющееся их произведением.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre class="synopsis">MatrixSum
(A)</pre><p>Вычисляет сумму всех элементов матрицы или вектора. То есть, складывает все элементы и возвращает
число, являющееся их суммой.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSu
mSquares</span></dt><dd><pre class="synopsis">MatrixSumSquares (A)</pre><p>Вычисляет сумму квадратов всех
элементов матрицы или вектора.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Returns a row vector of the indices of nonzero columns in the matrix <code
class="varname">M</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Returns a row vector of the indices of nonzero elements in the vector <code
class="varname">v</code>.</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct
(u,v)</pre><p>Get the outer product of two vectors. That is, suppose that
diff --git a/help/ru/html/ch11s09.html b/help/ru/html/ch11s09.html
index 2a4c2d1..e98eb05 100644
--- a/help/ru/html/ch11s09.html
+++ b/help/ru/html/ch11s09.html
@@ -50,7 +50,7 @@ result as a vector and not added together.</p></dd><dt><span class="term"><a nam
diagonal).
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multiplicities)</pre><p>Get the eigenvectors of a square matrix. Optionally get also
@@ -58,7 +58,7 @@ the eigenvalues and their algebraic multiplicities.
Currently only works for matrices of size up to 2 by 2.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Apply the Gram-Schmidt process (to the columns) with respect to
@@ -152,7 +152,7 @@ determinant.
of two matrices.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/KroneckerProduct"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre class="synopsis">LUDecomposition
(A, L, U)</pre><p>
@@ -182,7 +182,7 @@ determinant.
and <code class="varname">U</code> to <code class="constant">null</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/LU_decomposition";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Minor"></a>Minor</span></dt><dd><pre
class="synopsis">Minor (M,i,j)</pre><p>Get the <code class="varname">i</code>-<code class="varname">j</code>
minor of a matrix.</p><p>
@@ -218,7 +218,7 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<code class="varname">Q</code>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/QRDecomposition"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Return the Rayleigh quotient (also called the Rayleigh-Ritz
quotient or ratio) of a matrix and a vector.</p><p>
@@ -241,44 +241,44 @@ function of two arguments or it can be a matrix giving a sesquilinear form.
<a class="ulink" href="http://planetmath.org/SylvestersLaw"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.</p></dd><dt><span
class="term"><a name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(angle)</pre><p>Aliases: <code class="function">RotationMatrix</code></p><p>Return the matrix corresponding
to rotation around origin in R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
x-axis.</p></dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (angle)</pre><p>Return the matrix corresponding to rotation around origin in
R<sup>3</sup> about the
y-axis.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(angle)</pre><p>Return the matrix corresponding to rotation around origin in R<sup>3</sup> about the
z-axis.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Get a basis matrix for the rowspace of a matrix.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Evaluate (v,w) with respect to the sesquilinear form given
by the matrix A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunction</span></dt><dd><pre
class="synopsis">SesquilinearFormFunction (A)</pre><p>Return a function that evaluates two vectors with
respect to the sesquilinear form given by A.</p></dd><dt><span class="term"><a name="gel
-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returns the Smith normal form of a matrix over fields (will
end up with 1's on the diagonal).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Return the Smith normal form for square integer matrices
over integers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,args...)</pre><p>Solve linear system Mx=V, return solution V if there
is a unique solution, <code class="constant">null</code> otherwise. Extra two reference parameters can
optionally be used to get the reduced M and V.</p></dd><dt><span class="term"><a
name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre class="synopsis">ToeplitzMatrix (c,
r...)</pre><p>Return the Toeplitz matrix constructed given the first column c
and (optionally) the first row r. If only the column c is given then it is
conjugated and the nonconjugated version is used for the first row to give a
Hermitian matrix (if the first element is real of course).</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://planetmath.org/ToeplitzMatrix"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Trace"></a>Trace</span></dt><dd><pre
class="synopsis">Trace (M)</pre><p>Aliases: <code class="function">trace</code></p><p>Calculate the trace of
a matrix. That is the sum of the diagonal elements.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Trace"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre
class="synopsis">Transpose (M)</pre><p>Транспозиция матрицы. То же самое, что оператор <strong
class="userinput"><code>.'</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Aliases: <code class="function">vander</code></p><p>Return the
Vandermonde matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>The angle of two vectors with respect to inner product given by
<code class="varname">B</code>. If <code class="varname">B</code> is not given then the standard
Hermitian product is used. <code class="varname">B</code> can either be a sesquilinear
function of two arguments or it can be a matrix giving a sesquilinear form.
</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span></dt><dd><pre
class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>The direct sum of the vector spaces M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Intersection of the subspaces given by M and
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>The sum of the vector spaces M and N, that is {w | w=m+n, m
in M, n in N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Aliases: <code class="function">Adjugate</code></p><p>Get the classical
adjoint (adjugate) of a matrix.</p></dd><dt><span class="term"><a name="gel-function-cref"></a>cref</spa
n></dt><dd><pre class="synopsis">cref (M)</pre><p>Aliases: <code class="function">CREF</code> <code
class="function">ColumnReducedEchelonForm</code></p><p>Compute the Column Reduced Echelon
Form.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Aliases: <code class="function">Determinant</code></p><p>Get the determinant
of a matrix.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/Determinant2"; target="_top">Planetmath</a> for more
information.
</p></dd><dt><span class="term"><a name="gel-function-ref"></a>ref</span></dt><dd><pre
class="synopsis">ref (M)</pre><p>Aliases: <code class="function">REF</code> <code
class="function">RowEchelonForm</code></p><p>Get the row echelon form of a matrix. That is, apply gaussian
elimination but not backaddition to <code class="varname">M</code>. The pivot rows are
divided to make all pivots 1.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Row_echelon_form"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Aliases: <code class="function">RREF</code> <code
class="function">ReducedRowEchelonForm</code></p><p>Get the reduced row echelon form of a matrix. That is,
apply gaussian elimination together with backaddition to <code class="varname">M</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a>
for more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s08.html">Пред.</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s10.html">След.</a></td></tr><tr><td width="40%" align="left" valign="top">Операции с
матрицами </td><td width="20%" align="center"><a accesskey="h" href="index.html">Начало</a></td><td
width="40%" align="right" valign="top"> Комбинаторика</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s10.html b/help/ru/html/ch11s10.html
index f22cdd1..9e59f45 100644
--- a/help/ru/html/ch11s10.html
+++ b/help/ru/html/ch11s10.html
@@ -3,7 +3,10 @@
<a class="ulink" href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Get all combinations of k numbers from 1 to n as a vector of vectors.
(See also <a class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)
-</p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Двойной факториал: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
+</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre class="synopsis">DoubleFactorial
(n)</pre><p>Двойной факториал: <strong class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>
See
<a class="ulink" href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre
class="synopsis">Factorial (n)</pre><p>Факториал: <strong
class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>
@@ -14,17 +17,18 @@
<a class="ulink" href="http://planetmath.org/FallingFactorial"; target="_top">Planetmath</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre
class="synopsis">Fibonacci (x)</pre><p>Синонимы: <code class="function">fib</code></p><p>Вычисляет <code
class="varname">n</code>-ое число Фибоначчи. Это число, вычисляемое рекурсивно по формулам <strong
class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)</code></strong> и <strong
class="userinput"><code>Fibonacci(1) = Fibonacci(2) = 1</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a>
or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>
- Calculate the Frobenius number. That is calculate smallest
+ Calculate the Frobenius number. That is calculate largest
number that cannot be given as a non-negative integer linear
combination of a given vector of non-negative integers.
The vector can be given as separate numbers or a single vector.
All the numbers given should have GCD of 1.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(combining_rule)</pre><p>Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>
Find the vector <code class="varname">c</code> of non-negative integers
@@ -34,8 +38,18 @@
of non-negative integers.
</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.</p></dd><dt><span
class="term"><a name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coeffi
cients. Takes a vector of
+ </p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Aliases: <code class="function">HarmonicH</code></p><p>Harmonic Number, the <code
class="varname">n</code>th harmonic number of order <code class="varname">r</code>.
+ That is, it is the sum of <strong class="userinput"><code>1/k^r</code></strong> for <code
class="varname">k</code>
+ from 1 to n. Equivalent to <strong class="userinput"><code>sum k = 1 to n do
1/k^r</code></strong>.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a>
for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence";
target="_top">Wikipedia</a> for more information.
+ The sequence is <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 in
OEIS</a>.
+ </p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (seed_values,combining_rule,n)</pre><p>Compute linear recursive
sequence using Galois stepping.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Calculate multinomial coefficients. Takes a vector of
<code class="varname">k</code>
non-negative integers and computes the multinomial coefficient.
This corresponds to the coefficient in the homogeneous polynomial
@@ -51,7 +65,7 @@
<strong class="userinput"><code>Binomial(a+b,b)</code></strong>.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Multinomial_theorem";
target="_top">Wikipedia</a>,
<a class="ulink" href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/MultinomialCoefficient.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Get combination that would come after v in call to
@@ -71,6 +85,9 @@ do (
) while not IsNull(n:=NextCombination(n,6));</code></strong>
</pre><p>
See also <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Get the Pascal's triangle as a matrix. This will return
an <code class="varname">i</code>+1 by <code class="varname">i</code>+1 lower diagonal
matrix that is the Pascal's triangle after <code class="varname">i</code>
@@ -80,7 +97,7 @@ do (
</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Get all permutations of <code class="varname">k</code> numbers from 1 to <code
class="varname">n</code> as a vector of vectors.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Aliases: <code class="function">Pochhammer</code></p><p>(Pochhammer) Rising factorial: (n)_k =
n(n+1)...(n+(k-1)).</p><p>
See
<a class="ulink" href="http://planetmath.org/RisingFactorial"; target="_top">Planetmath</a> for
more information.
@@ -103,5 +120,5 @@ do (
<code class="varname">r</code> of numbers from 1 to <code class="varname">n</code>.</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a>
or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> for
more information.
</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Пред.</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">След.</a></td></tr><tr><td width="40%" align="left" valign="top">Линейная
алгебра </td><td width="20%" align="center"><a accesskey="h" href="index.html">Начало</a></td><td width="40%"
align="right" valign="top"> Calculus</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s11.html b/help/ru/html/ch11s11.html
index 1d4c83a..620b954 100644
--- a/help/ru/html/ch11s11.html
+++ b/help/ru/html/ch11s11.html
@@ -25,7 +25,7 @@ the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, whil
<strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Either <code class="varname">a</code>
or <code class="varname">b</code> can be <code class="constant">null</code>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(func,start,inc)</pre><p>Try to calculate an infinite product for a single parameter
function.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,inc)</pre><p>Try to calculate an infinite product for a
double parameter function with func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(func,start,inc)</pre><p>Try to calculate an infinite sum for a single parameter function.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,inc)</pre><p>Try to calculate an infinite sum for a double
parameter function with func(arg,n).</p></dd><d
t><span class="term"><a name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre
class="synopsis">IsContinuous (f,x0)</pre><p>Try and see if a real-valued function is continuous at x0 by
calculating the limit there.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Test for differentiability by approximating the left and
right limits and comparing.</p></dd><dt><span class="term"><a
name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Calculate the left limit of a real-valued function at x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Calculate the
limit of a real-valued function at x0. Tries to calculate both left and right limits.</p></dd><dt><span
class="term"><a name="gel-function-MidpointRule"></a>MidpointRule</sp
an></dt><dd><pre class="synopsis">MidpointRule (f,a,b,n)</pre><p>Integration by midpoint
rule.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Aliases: <code
class="function">NDerivative</code></p><p>Attempt to calculate numerical derivative.</p><p>
See
@@ -40,7 +40,7 @@ up to <code class="varname">N</code>th harmonic computed numerically. The coeff
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Return a function that is the Fourier series
of
<code class="function">f</code> with half-period <code class="varname">L</code> (that is defined
@@ -50,7 +50,7 @@ trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the cosine Fourier series of
@@ -65,7 +65,7 @@ Note that <strong class="userinput"><code>a@(1)</code></strong> is
the constant coefficient! That is, <strong class="userinput"><code>a@(n)</code></strong> refers to
the term <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Return a function that is the cosine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -76,7 +76,7 @@ only has cosine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierCosineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Return a vector of coefficients of
the sine Fourier series of
@@ -88,7 +88,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Return a function that is the sine
Fourier series of
<code class="function">f</code> with half-period <code class="varname">L</code>. That is,
@@ -99,7 +99,7 @@ only has sine terms. The series is computed up to the
computed by numerical integration using
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a>
or
<a class="ulink" href="http://mathworld.wolfram.com/FourierSineSeries.html";
target="_top">Mathworld</a> for more information.
</p><p>Version 1.0.7 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration by rule set in NumericalIntegralFunction of f
from a to b using NumericalIntegralSteps steps.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Attempt to calculate numerical left
derivative.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)</pre><p>Attempt to
calculate the limit of f(step_fun(i)) as i goes from 1 to N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">Nume
ricalRightDerivative (f,x0)</pre><p>Attempt to calculate numerical right derivative.</p></dd><dt><span
class="term"><a name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Return a function that is the odd periodic extension of
<code class="function">f</code> with half period <code class="varname">L</code>. That
diff --git a/help/ru/html/ch11s12.html b/help/ru/html/ch11s12.html
index 7f9a5d7..ec82f76 100644
--- a/help/ru/html/ch11s12.html
+++ b/help/ru/html/ch11s12.html
@@ -1,21 +1,21 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Functions</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Руководство пользователя Genius"><link rel="up" href="ch11.html" title="Глава 11. Список функций
GEL"><link rel="prev" href="ch11s11.html" title="Calculus"><link rel="next" href="ch11s13.html"
title="Решение уравнений"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Functions</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s11.html">Пред.</a> </td><th width="60%" align="center">Глава 11. Список функций GEL</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s13.html">След.</a></td></tr></table><hr></div><div cl
ass="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-functions"></a>Functions</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a name="gel-function-Argument"></a>Argument</span></dt><dd><pre
class="synopsis">Argument (z)</pre><p>Aliases: <code class="function">Arg</code> <code
class="function">arg</code></p><p>argument (angle) of complex number.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre class="synopsis">BesselJ0 (x)</pre><p>Bessel
function of the first kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1 (x)</pre><p>Bessel
function of the first kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn (n,x)</pre><p>Bessel
function of the first kind of order <code class="varname">n</code>. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0 (x)</pre><p>Bessel
function of the second kind of order 0. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1 (x)</pre><p>Bessel
function of the second kind of order 1. Only implemented for real numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn (n,x)</pre><p>Bessel
function of the second kind of order <code class="varname">n</code>. Only implemented for real
numbers.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.16 onwards.</p></dd><dt><span class="term"><a
name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre class="synopsis">DirichletKernel
(n,t)</pre><p>Dirichlet kernel of order <code class="varname">n</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre class="synopsis">DiscreteDelta
(v)</pre><p>Returns 1 if and only if all elements are zero.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Aliases: <code class="function">erf</code></p><p>The error function, 2/sqrt(pi) * int_0^x
e^(-t^2) dt.</p><p>
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
or
@@ -27,7 +27,7 @@
</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Aliases: <code class="function">Gamma</code></p><p>The Gamma function. Currently only
implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/GammaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returns 1 if and only if all elements are equal.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW (x)</pre><p>
The principal branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>.
@@ -38,7 +38,7 @@
See <a class="link" href="ch11s12.html#gel-function-LambertWm1"><code
class="function">LambertWm1</code></a> for the other real branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1 (x)</pre><p>
The minus-one branch of Lambert W function computed for only
real values greater than or equal to <strong class="userinput"><code>-1/e</code></strong>
@@ -48,32 +48,37 @@
See <a class="link" href="ch11s12.html#gel-function-LambertW"><code
class="function">LambertW</code></a> for the principal branch.
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (func,x,incr)</pre><p>Find the first value where f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Moebius mapping of the disk to itself mapping a to 0.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity
respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Moebius mapping using the cross ratio taking
infinity to infinity and z2,z3 to 1 and 0 respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 1 and z3,z4 to 0 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Moebius mapping using the cross ratio taking
infinity to 0 and z2,z4 to 1 and infinity respectively.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poisson kernel on D(0,R) (not normalized to
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Aliases: <code class="function">zeta</code></p><p>The Riemann zeta
function. Currently only implemented for real values.</p><p>
See
<a class="ulink" href="http://planetmath.org/RiemannZetaFunction"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>The unit step function is 0 for x<0, 1 otherwise. This is the
integral of the Dirac Delta function. Also called the Heaviside function.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> for
more information.
</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p>Функция <code class="function">cis</code>, то же самое, что <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Преобразует
градусы в радианы.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Преобразует
радианы в градусы.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Calculates the unnormalized sinc function, that is
<strong class="userinput"><code>sin(x)/x</code></strong>.
If you want the normalized function call <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> for more
information.
</p><p>Version 1.0.16 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Пред.</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s13.html">След.</a></td></tr><tr><td width="40%" align="left" valign="top">Calculus </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Начало</a></td><td width="40%" align="right"
valign="top"> Решение уравнений</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s13.html b/help/ru/html/ch11s13.html
index 7bca08b..d6dd86a 100644
--- a/help/ru/html/ch11s13.html
+++ b/help/ru/html/ch11s13.html
@@ -10,7 +10,7 @@
See
<a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
@@ -29,12 +29,12 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
@@ -73,7 +73,7 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.10 onwards.</p></dd><dt><span class="term"><a
name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Find root of a function using the bisection method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess
interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong
class="userinput"><code>f(b)</code></strong> should have opposite signs.
@@ -102,7 +102,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre class="synopsis">NewtonsMethod
(f,df,guess,epsilon,maxn)</pre><p>Find zeros using Newton's method. <code class="varname">f</code> is
the function and <code class="varname">df</code> is the derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
@@ -116,7 +116,7 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p><p>Version 1.0.18 onwards.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>
Compute roots of a polynomial (degrees 1 through 4)
using one of the formulas for such polynomials.
@@ -139,8 +139,9 @@
Returns a column vector of the two solutions.
</p><p>
See
- <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a> or
- <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> for more information.
+ <a class="ulink" href="http://planetmath.org/QuadraticFormula"; target="_top">Planetmath</a>, or
+ <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a>, or
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>
Compute roots of a quartic (degree 4) polynomial using the
quartic formula. The polynomial should be given as a
@@ -152,7 +153,7 @@
See
<a class="ulink" href="http://planetmath.org/QuarticFormula"; target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a>, or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
@@ -168,14 +169,14 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
- returns a 2 by <strong class="userinput"><code>n+1</code></strong> matrix with the
+ returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
@@ -209,5 +210,5 @@
</p><p>
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> or
- <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> for more information.
</p><p>Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s12.html">Пред.</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s14.html">След.</a></td></tr><tr><td width="40%" align="left" valign="top">Functions </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Начало</a></td><td width="40%" align="right"
valign="top"> Статистика</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s14.html b/help/ru/html/ch11s14.html
index e7db3d9..b977a41 100644
--- a/help/ru/html/ch11s14.html
+++ b/help/ru/html/ch11s14.html
@@ -1,14 +1,27 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Статистика</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Руководство пользователя Genius"><link rel="up"
href="ch11.html" title="Глава 11. Список функций GEL"><link rel="prev" href="ch11s13.html" title="Решение
уравнений"><link rel="next" href="ch11s15.html" title="Многочлены"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Статистика</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Пред.</a> </td><th width="60%" align="center">Глава 11. Список функций
GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch11s15.html">След.</a><
/td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-function-list-statistics"></a>Статистика</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Псевдонимы:
<code class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Вычисляет среднее арифметическое всех элементов матрицы.</p><p>Для
дополнительной информации смотрите <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a>.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral of the
GaussFunction from 0 to <code class="varname">x</code> (area under the normal curve).</p><p>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Статистика</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Руководство пользователя Genius"><link rel="up"
href="ch11.html" title="Глава 11. Список функций GEL"><link rel="prev" href="ch11s13.html" title="Решение
уравнений"><link rel="next" href="ch11s15.html" title="Многочлены"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Статистика</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s13.html">Пред.</a> </td><th width="60%" align="center">Глава 11. Список функций
GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch11s15.html">След.</a><
/td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title"
style="clear: both"><a name="genius-gel-function-list-statistics"></a>Статистика</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Псевдонимы:
<code class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Calculate average (the arithmetic mean) of an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral of the GaussFunction from 0 to <code
class="varname">x</code> (area under the normal curve).</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>The normalized Gauss distribution function (the normal curve).</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Normal_distribution";
target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Aliases: <code class="function">median</code></p><p>Calculate median of
an entire matrix.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
- </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix.</p><p>Для дополнительной информации смотрите <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a>.</p></dd><dt><span
class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre class="synopsis">RowMedian
(m)</pre><p>Calculate median of each row in a matrix and return a column
+ </p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">stdevp</code></p><p>Calculate the population standard deviation of a whole
matrix.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Aliases: <code class="function">RowMean</code></p><p>Calculate
average of each row in a matrix. That is, compute the
+ arithmetic mean.</p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> or
+ <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> for more information.
+ </p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre
class="synopsis">RowMedian (m)</pre><p>Calculate median of each row in a matrix and return a column
vector of the medians.</p><p>
See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> or
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html";
target="_top">Mathworld</a> for more information.
</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdevp</code></p><p>Calculate the population standard deviations of rows of a matrix and
return a vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Aliases: <code
class="function">rowstdev</code></p><p>Calculate the standard deviations of rows of a matrix and return a
vertical vector.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Aliases: <code class="function">stdev</code></p><p>Calculate
the standard deviation of a whole matrix.</p></dd></dl></div></div><div class="navfooter"><hr><table widt
h="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Пред.</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">След.</a></td></tr><tr><td width="40%" align="left" valign="top">Решение уравнений
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Начало</a></td><td width="40%"
align="right" valign="top"> Многочлены</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s15.html b/help/ru/html/ch11s15.html
index 62eedfe..3ff4c90 100644
--- a/help/ru/html/ch11s15.html
+++ b/help/ru/html/ch11s15.html
@@ -17,5 +17,5 @@
</pre><p>
</p><p>
See
- <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
+ <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
for more information.
</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Находит вторую производную многочлена (как
вектора).</p></dd><dt><span class="term"><a
name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre class="synopsis">PolyDerivative
(p)</pre><p>Находит производную многочлена (как вектора).</p></dd><dt><span class="term"><a
name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre class="synopsis">PolyToFunction
(p)</pre><p>Make function out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre class="synopsis">PolyToString
(p,var...)</pre><p>Make string out of a polynomial (as vector).</p></dd><dt><span class="term"><a
name="gel-function-SubtractPoly"></a>SubtractPoly</span></dt
<dd><pre class="synopsis">SubtractPoly (p1,p2)</pre><p>Subtract two polynomials (as
vectors).</p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Trim zeros from a polynomial (as
vector).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s14.html">Пред.</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Наверх</a></td><td width="40%" align="right">
<a accesskey="n" href="ch11s16.html">След.</a></td></tr><tr><td width="40%" align="left"
valign="top">Статистика </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Начало</a></td><td width="40%" align="right" valign="top"> Теория
множеств</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s18.html b/help/ru/html/ch11s18.html
index 9f56fb4..54d7009 100644
--- a/help/ru/html/ch11s18.html
+++ b/help/ru/html/ch11s18.html
@@ -1 +1,45 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Прочие</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Руководство пользователя Genius"><link rel="up" href="ch11.html" title="Глава 11. Список функций
GEL"><link rel="prev" href="ch11s17.html" title="Commutative Algebra"><link rel="next" href="ch11s19.html"
title="Symbolic Operations"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Прочие</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Пред.</a> </td><th width="60%" align="center">Глава 11. Список функций GEL</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s19.html">След.</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Прочие</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Преобразует вектор ASCII-значений в строку.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Преобразует вектор значений, представляющих собой
позиции букв в строке алфавита (начиная с 0), в строку.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Преобразует строк�
� в вектор ASCII-значений.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Преобразует строку в вектор значений, представляющих
собой позиции букв в строке алфавита (начиная с 0). Для неизвестных букв значения равны
-1.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Пред.</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">След.</a></td></tr><tr><td width="40%" align="left"
valign="top">Commutative Algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Начал�
�</a></td><td width="40%" align="right" valign="top"> Symbolic
Operations</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Прочие</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html"
title="Руководство пользователя Genius"><link rel="up" href="ch11.html" title="Глава 11. Список функций
GEL"><link rel="prev" href="ch11s17.html" title="Commutative Algebra"><link rel="next" href="ch11s19.html"
title="Symbolic Operations"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Прочие</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Пред.</a> </td><th width="60%" align="center">Глава 11. Список функций GEL</th><td
width="20%" align="right"> <a accesskey="n" href="ch11s19.html">След.</a></td></tr></table><hr></div><div
class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Прочие</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vec)</pre><p>Convert a vector of ASCII values to a string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vec,alphabet)</pre><p>Convert a vector of 0-based alphabet values
(positions in the alphabet string) to a string. A <code class="constant">null</code> vector results in an
empty string.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre><p>
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Convert a string to a (row) vector of ASCII values.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.
+ </p><p>
+ Example:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>
+ </p><p>
+ See
+ <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> for more
information.
+ </p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alphabet)</pre><p>Convert a string to a (row) vector of 0-based
alphabet values
+ (positions in the alphabet string), -1's for unknown letters.
+ An empty string results in a <code class="constant">null</code>.
+ See also
+ <a class="link" href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.
+ </p><p>
+ Examples:
+ </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre><p>
+ </p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Пред.</a> </td><td width="20%"
align="center"><a accesskey="u" href="ch11.html">Наверх</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">След.</a></td></tr><tr><td width="40%" align="left"
valign="top">Commutative Algebra </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Начало</a></td><td width="40%" align="right" valign="top"> Symbolic
Operations</td></tr></table></div></body></html>
diff --git a/help/ru/html/ch11s20.html b/help/ru/html/ch11s20.html
index 93ee0e7..70013a3 100644
--- a/help/ru/html/ch11s20.html
+++ b/help/ru/html/ch11s20.html
@@ -102,7 +102,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
</pre><p>
@@ -153,7 +153,7 @@ optionally the limits as <strong class="userinput"><code>x1,x2,y1,y2</code></str
Examples:
</p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","The
Solution")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","The
Solution")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","The
7th roots of unity")</code></strong>
</pre><p>
@@ -330,7 +330,7 @@ limits as <strong class="userinput"><code>x1,x2,y1,y2</code></strong>.
<code class="varname">n</code> by 3 matrix for a longer polyline.
</p><p>
Extra parameters can be added to specify line color, thickness,
- arrows, the plotting window, or legend.
+ the plotting window, or legend.
You can do this by adding an argument string <strong
class="userinput"><code>"color"</code></strong>,
<strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>,
diff --git a/help/ru/html/index.html b/help/ru/html/index.html
index 3b67b65..16bb335 100644
--- a/help/ru/html/index.html
+++ b/help/ru/html/index.html
@@ -58,7 +58,7 @@
EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED OF
THE POSSIBILITY OF SUCH DAMAGES.
</p></li></ol></div><p>
- </p></div></div><div><div class="legalnotice"><a name="idm45495306371328"></a><p
class="legalnotice-title"><b>Обратная связь</b></p><p>
+ </p></div></div><div><div class="legalnotice"><a name="idm51"></a><p
class="legalnotice-title"><b>Обратная связь</b></p><p>
To report a bug or make a suggestion regarding the <span class="application">Genius Mathematics
Tool</span>
application or this manual, please visit the
<a class="ulink" href="http://www.jirka.org/genius.html"; target="_top">Genius
diff --git a/help/sv/html/ch02s02.html b/help/sv/html/ch02s02.html
index fda2678..5859f59 100644
--- a/help/sv/html/ch02s02.html
+++ b/help/sv/html/ch02s02.html
@@ -1 +1 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Då du startar
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Handbok för Genius"><link rel="up" href="ch02.html" title="Kapitel 2. Komma
igång"><link rel="prev" href="ch02.html" title="Kapitel 2. Komma igång"><link rel="next" href="ch03.html"
title="Kapitel 3. Grundläggande användning"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Då du startar Genius</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch02.html">Föregående</a> </td><th width="60%" align="center">Kapitel 2. Komma
igång</th><td width="20%" align="right"> <a accesskey="n"
href="ch03.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
c
lass="title" style="clear: both"><a name="genius-when-start"></a>Då du startar
Genius</h2></div></div></div><p>Då du startar GNOME-versionen av <span class="application">Genius
matematikverktyg</span> kommer fönstret som avbildas i <a class="xref" href="ch02s02.html#mainwindow-fig"
title="Figur 2.1. Genius matematikverktyg-fönstret">Figur 2.1, ”<span class="application">Genius
matematikverktyg</span>-fönstret”</a> att visas.</p><div class="figure"><a name="mainwindow-fig"></a><p
class="title"><b>Figur 2.1. <span class="application">Genius matematikverktyg</span>-fönstret</b></p><div
class="figure-contents"><div class="screenshot"><div class="mediaobject"><img src="figures/genius_window.png"
alt="Visar huvudfönster för Genius matematikverktyg. Innehåller titelrad, menyrad, verktygsfält och
arbetsyta. Menyraden innehåller menyerna Arkiv, Redigera, Miniräknare, Exempel, Program, Inställningar och
Hjälp."></div></div></div></div><br class="figure-break"><p>Fö
nstret för <span class="application">Genius matematikverktyg</span> innehåller följande element:</p><div
class="variablelist"><dl class="variablelist"><dt><span class="term">Menyrad.</span></dt><dd><p>Menyerna på
menyraden innehåller alla kommandon som du behöver för att arbeta med filer i <span
class="application">Genius matematikverktyg</span>.<span class="guilabel">Arkiv</span>-menyn innehåller
poster för att läsa in och spara objekt och skapa nya program. Kommandot <span class="guilabel">Läs in och
kör...</span> öppnar inte ett nytt fönster för programmet, utan kör bara programmet direkt. Det är ekvivalent
med kommandot <span class="command"><strong>läs in</strong></span>.</p><p>Menyn <span
class="guilabel">Miniräknare</span> kontrollerar miniräknarmotorn. Den låter dig välja det aktuellt valda
programmet eller att avbryta den pågående beräkningen. Du kan också se det fulla uttrycket för det senaste
svaret (praktiskt om det senaste svaret var f
ör stort för att passa i konsolen), eller så kan du se en lista över värdena för alla användardefinierade
variabler. Du kan också övervaka användarvariabler, vilket är särskilt användbart under tiden en lång
beräkning pågår, eller för att felsöka ett specifikt program. Slutligen låter <span
class="guilabel">Miniräknare</span> dig att rita funktionsgrafer med en användarvänlig
dialogruta.</p><p>Menyn <span class="guilabel">Exempel</span> är en lista över exempelprogram eller
demonstrationer. Om du öppnar menyn kommer den läsa in exemplet i ett nytt program vilket du kan köra,
redigera, ändra och spara. Dessa program bör vara väl dokumenterade och demonstrerar allmänt antingen någon
funktion i <span class="application">Genius matematikverktyg</span> eller något matematiskt
koncept.</p><p>Menyn <span class="guilabel">Program</span> listar aktuellt öppna program och låter dig växla
mellan dem.</p><p>De andra menyerna har samma bekanta funktio
ner som i andra program.</p></dd><dt><span class="term">Verktygsfält.</span></dt><dd><p>Verktygsfältet
innehåller en delmängd av kommandona du kan komma åt från menyraden.</p></dd><dt><span
class="term">Arbetsyta</span></dt><dd><p>Arbetsytan är den primära metoden för att interagera med
programmet.</p><p>Arbetsytan har ursprungligen bara fliken <span class="guilabel">Konsol</span>, vilken är
huvudsättet att interagera med miniräknaren. Här skriver du in uttryck och resultaten visas omedelbart efter
att du tryckt på Returknappen.</p><p>Alternativt kan du skriva längre program och de kan visas i separata
flikar. Programmen är en uppsättning kommandon eller funktioner som kan köras alla på en gång snarare mata in
dem i kommandoraden. Programmen kan sparas i filer för senare användning.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch02.html">Föreg
ående</a> </td><td width="20%" align="center"><a accesskey="u" href="ch02.html">Upp</a></td><td width="40%"
align="right"> <a accesskey="n" href="ch03.html">Nästa</a></td></tr><tr><td width="40%" align="left"
valign="top">Kapitel 2. Komma igång </td><td width="20%" align="center"><a accesskey="h"
href="index.html">Hem</a></td><td width="40%" align="right" valign="top"> Kapitel 3. Grundläggande
användning</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Då du startar
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Handbok för Genius"><link rel="up" href="ch02.html" title="Kapitel 2. Komma
igång"><link rel="prev" href="ch02.html" title="Kapitel 2. Komma igång"><link rel="next" href="ch03.html"
title="Kapitel 3. Grundläggande användning"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Då du startar Genius</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch02.html">Föregående</a> </td><th width="60%" align="center">Kapitel 2. Komma
igång</th><td width="20%" align="right"> <a accesskey="n"
href="ch03.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2
c
lass="title" style="clear: both"><a name="genius-when-start"></a>Då du startar
Genius</h2></div></div></div><p>Då du startar GNOME-versionen av <span class="application">Genius
matematikverktyg</span> kommer fönstret som avbildas i <a class="xref" href="ch02s02.html#mainwindow-fig"
title="Figur 2.1. Genius matematikverktyg-fönstret">Figur 2.1, ”<span class="application">Genius
matematikverktyg</span>-fönstret”</a> att visas.</p><div class="figure"><a name="mainwindow-fig"></a><p
class="title"><b>Figur 2.1. <span class="application">Genius matematikverktyg</span>-fönstret</b></p><div
class="figure-contents"><div class="screenshot"><div class="mediaobject"><img src="figures/genius_window.png"
alt="Shows Genius Mathematics Tool main window. Contains titlebar, menubar, toolbar and working area. Menubar
contains Arkiv, Redigera, Miniräknare, Exempel, Program, Inställningar, and Hjälp
menus."></div></div></div></div><br class="figure-break"><p>Fönstret för <span cl
ass="application">Genius matematikverktyg</span> innehåller följande element:</p><div
class="variablelist"><dl class="variablelist"><dt><span class="term">Menyrad.</span></dt><dd><p>Menyerna på
menyraden innehåller alla kommandon som du behöver för att arbeta med filer i <span
class="application">Genius matematikverktyg</span>.<span class="guilabel">Arkiv</span>-menyn innehåller
poster för att läsa in och spara objekt och skapa nya program. Kommandot <span class="guilabel">Läs in och
kör...</span> öppnar inte ett nytt fönster för programmet, utan kör bara programmet direkt. Det är ekvivalent
med kommandot <span class="command"><strong>läs in</strong></span>.</p><p>Menyn <span
class="guilabel">Miniräknare</span> kontrollerar miniräknarmotorn. Den låter dig välja det aktuellt valda
programmet eller att avbryta den pågående beräkningen. Du kan också se det fulla uttrycket för det senaste
svaret (praktiskt om det senaste svaret var för stort för att p
assa i konsolen), eller så kan du se en lista över värdena för alla användardefinierade variabler. Du kan
också övervaka användarvariabler, vilket är särskilt användbart under tiden en lång beräkning pågår, eller
för att felsöka ett specifikt program. Slutligen låter <span class="guilabel">Miniräknare</span> dig att rita
funktionsgrafer med en användarvänlig dialogruta.</p><p>Menyn <span class="guilabel">Exempel</span> är en
lista över exempelprogram eller demonstrationer. Om du öppnar menyn kommer den läsa in exemplet i ett nytt
program vilket du kan köra, redigera, ändra och spara. Dessa program bör vara väl dokumenterade och
demonstrerar allmänt antingen någon funktion i <span class="application">Genius matematikverktyg</span> eller
något matematiskt koncept.</p><p>Menyn <span class="guilabel">Program</span> listar aktuellt öppna program
och låter dig växla mellan dem.</p><p>De andra menyerna har samma bekanta funktioner som i andra prog
ram.</p></dd><dt><span class="term">Verktygsfält.</span></dt><dd><p>Verktygsfältet innehåller en delmängd av
kommandona du kan komma åt från menyraden.</p></dd><dt><span
class="term">Arbetsyta</span></dt><dd><p>Arbetsytan är den primära metoden för att interagera med
programmet.</p><p>Arbetsytan har ursprungligen bara fliken <span class="guilabel">Konsol</span>, vilken är
huvudsättet att interagera med miniräknaren. Här skriver du in uttryck och resultaten visas omedelbart efter
att du tryckt på Returknappen.</p><p>Alternativt kan du skriva längre program och de kan visas i separata
flikar. Programmen är en uppsättning kommandon eller funktioner som kan köras alla på en gång snarare mata in
dem i kommandoraden. Programmen kan sparas i filer för senare användning.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch02.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch02.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch03.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Kapitel 2.
Komma igång </td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td
width="40%" align="right" valign="top"> Kapitel 3. Grundläggande
användning</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch05s07.html b/help/sv/html/ch05s07.html
index 78e3b26..3cc605e 100644
--- a/help/sv/html/ch05s07.html
+++ b/help/sv/html/ch05s07.html
@@ -1,6 +1,6 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Lista över
GEL-operatorer</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Handbok för Genius"><link rel="up" href="ch05.html" title="Kapitel 5. Grunderna i
GEL"><link rel="prev" href="ch05s06.html" title="Moduloberäkning"><link rel="next" href="ch06.html"
title="Kapitel 6. Programmering med GEL"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Lista över GEL-operatorer</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch05s06.html">Föregående</a> </td><th width="60%" align="center">Kapitel
5. Grunderna i GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div>
<div><h2 class="title" style="clear: both"><a name="genius-gel-operator-list"></a>Lista över
GEL-operatorer</h2></div></div></div><p>Allt i GEL är bara ett uttryck. Uttryck slås samman med olika
operatorer. Som vi har sett är till och med avskiljaren helt enkelt en binär operator i GEL. Här är en lista
över operatorerna i GEL.</p><div class="variablelist"><dl class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>Avskiljaren, evaluerar helt enkelt både <code
class="varname">a</code> och <code class="varname">b</code>, men returnerar bara resultatet av <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>Tilldelningsoperatorn. Denna tilldelar <code
class="varname">b</code> till <code class="varname">a</code> (<code class="varname">a</code> måste vara ett
giltigt <a class="link" href="ch06s09.html" title="Vvärden">vvärde</a>) (o
bservera dock att denna operator kan översättas till <code class="literal">==</code> om den används där ett
booleskt uttryck förväntas)</p></dd><dt><span class="term"><strong
class="userinput"><code>a:=b</code></strong></span></dt><dd><p>Tilldelningsoperatorn. Tilldelar <code
class="varname">b</code> till <code class="varname">a</code> (<code class="varname">a</code> måste vara ett
giltigt <a class="link" href="ch06s09.html" title="Vvärden">vvärde</a>) Detta skiljer sig från <code
class="literal">=</code> eftersom det aldrig översätts till <code
class="literal">==</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>|a|</code></strong></span></dt><dd><p>Absolutbelopp. Om uttrycket är ett komplext tak
kommer resultatet vara avståndet från origo. Till exempel: <strong class="userinput"><code>|3 *
e^(1i*pi)|</code></strong> returnerar 3.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld</a>
för mer information.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Exponentiering, upphöjer <code
class="varname">a</code> till exponenten <code class="varname">b</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Elementvis
exponentiering. Upphöj varje element i en matris <code class="varname">a</code> till exponenten <code
class="varname">b</code>. Eller om <code class="varname">b</code> är en matris med samma storlek som <code
class="varname">a</code>, gör i så fall operationen elementvis. Om <code class="varname">a</code> är ett tal
och <code class="varname">b</code> är en matris så skapar det en matris av samma storlek som <code
class="varname">b</code> med <code class="varname">a</code> upphöjt till alla de olika exponenterna i <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a+b</code></strong></span>
</dt><dd><p>Addition. Adderar två tal, matriser, funktioner eller strängar. Om du lägger till en sträng
kommer resultatet att vara en sträng. Om en är en kvadratisk matris och den andra ett tal kommer talet att
multipliceras med identitetsmatrisen.</p></dd><dt><span class="term"><strong
class="userinput"><code>a-b</code></strong></span></dt><dd><p>Subtraktion. Subtrahera två tal, matriser eller
funktioner.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Multiplikation. Detta är vanlig
matrismultiplikation.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Elementvis multiplikation om <code
class="varname">a</code> och <code class="varname">b</code> är matriser.</p></dd><dt><span
class="term"><strong class="userinput"><code>a/b</code></strong></span></dt><dd><p>Division. Då <code
class="varname">a</code> och <code class="varname">b</code> bara är tal är detta v
anlig division. Då de är matriser är detta ekvivalent med <strong
class="userinput"><code>a*b^-1</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>Elementvis division. Samma som <strong
class="userinput"><code>a/b</code></strong> för tal, men opererar elementvis på matriser.</p></dd><dt><span
class="term"><strong class="userinput"><code>a\b</code></strong></span></dt><dd><p>Baklängesdivision. Det
vill säga detta är samma sak som <strong class="userinput"><code>b/a</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Elementvis
baklängesdivision.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>Modulooperatorn. Detta slår inte på <a
class="link" href="ch05s06.html" title="Moduloberäkning">moduloläget</a>, utan returnerar bara resten av
<strong class="userinput"><code>a/b</code></
strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>Elementvis modulooperator. Returnerar resten
efter elementvis <strong class="userinput"><code>a./b</code></strong> av heltal.</p></dd><dt><span
class="term"><strong class="userinput"><code>a mod b</code></strong></span></dt><dd><p>Modulär
evalueringsoperator. Uttrycket <code class="varname">a</code> evalueras modulo <code
class="varname">b</code>. Se <a class="xref" href="ch05s06.html"
title="Moduloberäkning">”Moduloberäkning”</a>. Vissa funktioner och operatorer beter sig annorlunda modulo
ett heltal.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Fakultetsoperator. Detta är som <strong
class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Semifakultetsoperator. Detta är som <strong
class="userinpu
t"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a==b</code></strong></span></dt><dd><p>Likhetsoperator. Returnerar <code
class="constant">true</code> eller <code class="constant">false</code> beroende på om <code
class="varname">a</code> och <code class="varname">b</code> är lika eller inte.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Olikhetsoperator,
returnerar <code class="constant">true</code> om <code class="varname">a</code> inte är lika med <code
class="varname">b</code>, returnerar annars <code class="constant">false</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Alternativ
olikhetsoperator, returnerar <code class="constant">true</code> om <code class="varname">a</code> inte är
lika med <code class="varname">b</code>, returnerar annars <code class="constant">false</code>.</p></d
d><dt><span class="term"><strong class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Mindre än
eller lika med-operator, returnerar <code class="constant">true</code> om <code class="varname">a</code> är
mindre eller lika med <code class="varname">b</code>, returnerar annars <code class="constant">false</code>.
Dessa kan kombineras som i <strong class="userinput"><code>a <= b <= c</code></strong> (kan också
kombineras med mindre än-operatorn).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>Större än eller lika med-operator,
returnerar <code class="constant">true</code> om <code class="varname">a</code> är större eller lika med
<code class="varname">b</code>, returnerar annars <code class="constant">false</code>. Dessa kan kombineras
som i <strong class="userinput"><code>a >= b >= c</code></strong> (kan också kombineras med större
än-operatorn).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<b</code></strong></span></dt><dd><p>Mindre än-operator, returnerar <code
class="constant">true</code> om <code class="varname">a</code> är mindre än <code class="varname">b</code>,
returnerar annars <code class="constant">false</code>. Dessa kan kombineras som i <strong
class="userinput"><code>a < b < c</code></strong> (kan också kombineras med mindre än eller lika
med-operatorn).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>Större än-operator, returnerar <code
class="constant">true</code> om <code class="varname">a</code> är större än <code class="varname">b</code>,
returnerar annars <code class="constant">false</code>. Dessa kan kombineras som i <strong
class="userinput"><code>a > b > c</code></strong> (kan också kombineras med större än eller lika
med-operatorn).</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=>b</code></strong></span><
/dt><dd><p>Jämförelseoperator. Om <code class="varname">a</code> är lika med <code class="varname">b</code>
returnerar den 0, om <code class="varname">a</code> är mindre än <code class="varname">b</code> returnerar
den -1 och om <code class="varname">a</code> är större än <code class="varname">b</code> returnerar den
1.</p></dd><dt><span class="term"><strong class="userinput"><code>a and
b</code></strong></span></dt><dd><p>Logiskt och. Returnerar true om både <code class="varname">a</code> och
<code class="varname">b</code> är true, returnerar annars false. Om tal gives behandlas nollskilda tal som
true.</p></dd><dt><span class="term"><strong class="userinput"><code>a or
b</code></strong></span></dt><dd><p>Logiskt eller. Returnerar true om antingen <code class="varname">a</code>
eller <code class="varname">b</code> är true, returnerar annars false. Om tal gives behandlas nollskilda tal
som true.</p></dd><dt><span class="term"><strong class="userinput"><code>a xor
b</code></strong></span></dt><dd><p>Logiskt uteslutande eller. Returnerar true om exakt en av <code
class="varname">a</code> eller <code class="varname">b</code> är true, returnerar annars false. Om tal gives
behandlas nollskilda tal som true.</p></dd><dt><span class="term"><strong class="userinput"><code>not
a</code></strong></span></dt><dd><p>Logiskt inte. Returnerar den logiska negationen till <code
class="varname">a</code></p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>Negationsoperator. Returnerar negativet av ett
tal eller en matris (arbetar elementvis på en matris).</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Variabelreferens (för att skicka en referens
till en variabel). Se <a class="xref" href="ch06s08.html"
title="Referenser">”Referenser”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></strong></span></dt><dd><p>Varia
beldereferering (för att komma åt en refererad variabel). Se <a class="xref" href="ch06s08.html"
title="Referenser">”Referenser”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Konjugattransponatet för matris. Det vill säga
rader och kolumner byter plats och vi tar komplexkonjugatet av alla poster. Det vill säga om i,j-elementet av
<code class="varname">a</code> är x+iy så är j,i-elementet av <strong
class="userinput"><code>a'</code></strong> då x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Matristransponat, konjugerar inte posterna. Det
vill säga i,j-elementet av <code class="varname">a</code> blir j,i-elementet av <strong
class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>Hämta element för en matris i rad <code
class="varname">b</code> och kolu
mn <code class="varname">c</code>. Om <code class="varname">b</code>, <code class="varname">c</code> är
vektorer så ger detta de motsvarande raderna, kolumnerna eller delmatriserna.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Hämta rad av en matris
(eller flera rader om <code class="varname">b</code> är en vektor).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Samma som ovan.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Hämta kolumn av en
matris (eller flera kolumner om <code class="varname">b</code> är en vektor).</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Samma som
ovan.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Hämta ett element från en matris behandlad
som en vektor.
Detta kommer traversera matrisen radvis.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Bygg en vektor från <code
class="varname">a</code> till <code class="varname">b</code> (eller ange en rad- och kolumnregion för <code
class="literal">@</code>-operatorn). Till exempel kan vi för att få raderna 2 till 4 av matrisen <code
class="varname">A</code> göra </p><pre class="programlisting">A@(2:4,)
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Lista över
GEL-operatorer</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
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header"><tr><th colspan="3" align="center">Lista över GEL-operatorer</th></tr><tr><td width="20%"
align="left"><a accesskey="p" href="ch05s06.html">Föregående</a> </td><th width="60%" align="center">Kapitel
5. Grunderna i GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div>
<div><h2 class="title" style="clear: both"><a name="genius-gel-operator-list"></a>Lista över
GEL-operatorer</h2></div></div></div><p>Allt i GEL är bara ett uttryck. Uttryck slås samman med olika
operatorer. Som vi har sett är till och med avskiljaren helt enkelt en binär operator i GEL. Här är en lista
över operatorerna i GEL.</p><div class="variablelist"><dl class="variablelist"><dt><span class="term"><strong
class="userinput"><code>a;b</code></strong></span></dt><dd><p>Avskiljaren, evaluerar helt enkelt både <code
class="varname">a</code> och <code class="varname">b</code>, men returnerar bara resultatet av <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a=b</code></strong></span></dt><dd><p>Tilldelningsoperatorn. Denna tilldelar <code
class="varname">b</code> till <code class="varname">a</code> (<code class="varname">a</code> måste vara ett
giltigt <a class="link" href="ch06s09.html" title="Vvärden">vvärde</a>) (o
bservera dock att denna operator kan översättas till <code class="literal">==</code> om den används där ett
booleskt uttryck förväntas)</p></dd><dt><span class="term"><strong
class="userinput"><code>a:=b</code></strong></span></dt><dd><p>Tilldelningsoperatorn. Tilldelar <code
class="varname">b</code> till <code class="varname">a</code> (<code class="varname">a</code> måste vara ett
giltigt <a class="link" href="ch06s09.html" title="Vvärden">vvärde</a>) Detta skiljer sig från <code
class="literal">=</code> eftersom det aldrig översätts till <code
class="literal">==</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>|a|</code></strong></span></dt><dd><p>Absolutbelopp. Om uttrycket är ett komplext tak
kommer resultatet vara avståndet från origo. Till exempel: <strong class="userinput"><code>|3 *
e^(1i*pi)|</code></strong> returnerar 3.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/AbsoluteValue.html"; target="_top">Mathworld</a>
för mer information.</p></dd><dt><span class="term"><strong
class="userinput"><code>a^b</code></strong></span></dt><dd><p>Exponentiering, upphöjer <code
class="varname">a</code> till exponenten <code class="varname">b</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.^b</code></strong></span></dt><dd><p>Elementvis
exponentiering. Upphöj varje element i en matris <code class="varname">a</code> till exponenten <code
class="varname">b</code>. Eller om <code class="varname">b</code> är en matris med samma storlek som <code
class="varname">a</code>, gör i så fall operationen elementvis. Om <code class="varname">a</code> är ett tal
och <code class="varname">b</code> är en matris så skapar det en matris av samma storlek som <code
class="varname">b</code> med <code class="varname">a</code> upphöjt till alla de olika exponenterna i <code
class="varname">b</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a+b</code></strong></span>
</dt><dd><p>Addition. Adderar två tal, matriser, funktioner eller strängar. Om du lägger till en sträng
kommer resultatet att vara en sträng. Om en är en kvadratisk matris och den andra ett tal kommer talet att
multipliceras med identitetsmatrisen.</p></dd><dt><span class="term"><strong
class="userinput"><code>a-b</code></strong></span></dt><dd><p>Subtraktion. Subtrahera två tal, matriser eller
funktioner.</p></dd><dt><span class="term"><strong
class="userinput"><code>a*b</code></strong></span></dt><dd><p>Multiplikation. Detta är vanlig
matrismultiplikation.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.*b</code></strong></span></dt><dd><p>Elementvis multiplikation om <code
class="varname">a</code> och <code class="varname">b</code> är matriser.</p></dd><dt><span
class="term"><strong class="userinput"><code>a/b</code></strong></span></dt><dd><p>Division. Då <code
class="varname">a</code> och <code class="varname">b</code> bara är tal är detta v
anlig division. Då de är matriser är detta ekvivalent med <strong
class="userinput"><code>a*b^-1</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a./b</code></strong></span></dt><dd><p>Elementvis division. Samma som <strong
class="userinput"><code>a/b</code></strong> för tal, men opererar elementvis på matriser.</p></dd><dt><span
class="term"><strong class="userinput"><code>a\b</code></strong></span></dt><dd><p>Baklängesdivision. Det
vill säga detta är samma sak som <strong class="userinput"><code>b/a</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a.\b</code></strong></span></dt><dd><p>Elementvis
baklängesdivision.</p></dd><dt><span class="term"><strong
class="userinput"><code>a%b</code></strong></span></dt><dd><p>Modulooperatorn. Detta slår inte på <a
class="link" href="ch05s06.html" title="Moduloberäkning">moduloläget</a>, utan returnerar bara resten av
heltalsdivisionen <strong class="userinput">
<code>a/b</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.%b</code></strong></span></dt><dd><p>Elementvis modulooperator. Returnerar resten
efter elementvis division <strong class="userinput"><code>a./b</code></strong> av heltal.</p></dd><dt><span
class="term"><strong class="userinput"><code>a mod b</code></strong></span></dt><dd><p>Modulär
evalueringsoperator. Uttrycket <code class="varname">a</code> evalueras modulo <code
class="varname">b</code>. Se <a class="xref" href="ch05s06.html"
title="Moduloberäkning">”Moduloberäkning”</a>. Vissa funktioner och operatorer beter sig annorlunda modulo
ett heltal.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!</code></strong></span></dt><dd><p>Fakultetsoperator. Detta är som <strong
class="userinput"><code>1*...*(n-2)*(n-1)*n</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a!!</code></strong></span></dt><dd><p>Semifakultetsoperator. Detta är
som <strong class="userinput"><code>1*...*(n-4)*(n-2)*n</code></strong>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a==b</code></strong></span></dt><dd><p>Likhetsoperator.
Returnerar <code class="constant">true</code> eller <code class="constant">false</code> beroende på om <code
class="varname">a</code> och <code class="varname">b</code> är lika eller inte.</p></dd><dt><span
class="term"><strong class="userinput"><code>a!=b</code></strong></span></dt><dd><p>Olikhetsoperator,
returnerar <code class="constant">true</code> om <code class="varname">a</code> inte är lika med <code
class="varname">b</code>, returnerar annars <code class="constant">false</code>.</p></dd><dt><span
class="term"><strong class="userinput"><code>a<>b</code></strong></span></dt><dd><p>Alternativ
olikhetsoperator, returnerar <code class="constant">true</code> om <code class="varname">a</code> inte är
lika med <code class="varname">b</code>, returnerar annars <code class="con
stant">false</code>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a<=b</code></strong></span></dt><dd><p>Mindre än eller lika med-operator,
returnerar <code class="constant">true</code> om <code class="varname">a</code> är mindre eller lika med
<code class="varname">b</code>, returnerar annars <code class="constant">false</code>. Dessa kan kombineras
som i <strong class="userinput"><code>a <= b <= c</code></strong> (kan också kombineras med mindre
än-operatorn).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>=b</code></strong></span></dt><dd><p>Större än eller lika med-operator,
returnerar <code class="constant">true</code> om <code class="varname">a</code> är större eller lika med
<code class="varname">b</code>, returnerar annars <code class="constant">false</code>. Dessa kan kombineras
som i <strong class="userinput"><code>a >= b >= c</code></strong> (kan också kombineras med större
än-operatorn).</p></dd><dt
<span class="term"><strong class="userinput"><code>a<b</code></strong></span></dt><dd><p>Mindre
än-operator, returnerar <code class="constant">true</code> om <code class="varname">a</code> är mindre än
<code class="varname">b</code>, returnerar annars <code class="constant">false</code>. Dessa kan kombineras
som i <strong class="userinput"><code>a < b < c</code></strong> (kan också kombineras med mindre än
eller lika med-operatorn).</p></dd><dt><span class="term"><strong
class="userinput"><code>a>b</code></strong></span></dt><dd><p>Större än-operator, returnerar <code
class="constant">true</code> om <code class="varname">a</code> är större än <code class="varname">b</code>,
returnerar annars <code class="constant">false</code>. Dessa kan kombineras som i <strong
class="userinput"><code>a > b > c</code></strong> (kan också kombineras med större än eller lika
med-operatorn).</p></dd><dt><span class="term"><strong class="userinput"><code>a<=&g
t;b</code></strong></span></dt><dd><p>Jämförelseoperator. Om <code class="varname">a</code> är lika med
<code class="varname">b</code> returnerar den 0, om <code class="varname">a</code> är mindre än <code
class="varname">b</code> returnerar den -1 och om <code class="varname">a</code> är större än <code
class="varname">b</code> returnerar den 1.</p></dd><dt><span class="term"><strong class="userinput"><code>a
and b</code></strong></span></dt><dd><p>Logiskt och. Returnerar true om både <code class="varname">a</code>
och <code class="varname">b</code> är true, returnerar annars false. Om tal gives behandlas nollskilda tal
som true.</p></dd><dt><span class="term"><strong class="userinput"><code>a or
b</code></strong></span></dt><dd><p>Logiskt eller. Returnerar true om antingen <code class="varname">a</code>
eller <code class="varname">b</code> är true, returnerar annars false. Om tal gives behandlas nollskilda tal
som true.</p></dd><dt><span class="term"><strong cla
ss="userinput"><code>a xor b</code></strong></span></dt><dd><p>Logiskt uteslutande eller. Returnerar true om
exakt en av <code class="varname">a</code> eller <code class="varname">b</code> är true, returnerar annars
false. Om tal gives behandlas nollskilda tal som true.</p></dd><dt><span class="term"><strong
class="userinput"><code>not a</code></strong></span></dt><dd><p>Logiskt inte. Returnerar den logiska
negationen till <code class="varname">a</code></p></dd><dt><span class="term"><strong
class="userinput"><code>-a</code></strong></span></dt><dd><p>Negationsoperator. Returnerar negativet av ett
tal eller en matris (arbetar elementvis på en matris).</p></dd><dt><span class="term"><strong
class="userinput"><code>&a</code></strong></span></dt><dd><p>Variabelreferens (för att skicka en referens
till en variabel). Se <a class="xref" href="ch06s08.html"
title="Referenser">”Referenser”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>*a</code></stro
ng></span></dt><dd><p>Variabeldereferering (för att komma åt en refererad variabel). Se <a class="xref"
href="ch06s08.html" title="Referenser">”Referenser”</a>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a'</code></strong></span></dt><dd><p>Konjugattransponatet för matris. Det vill säga
rader och kolumner byter plats och vi tar komplexkonjugatet av alla poster. Det vill säga om i,j-elementet av
<code class="varname">a</code> är x+iy så är j,i-elementet av <strong
class="userinput"><code>a'</code></strong> då x-iy.</p></dd><dt><span class="term"><strong
class="userinput"><code>a.'</code></strong></span></dt><dd><p>Matristransponat, konjugerar inte posterna. Det
vill säga i,j-elementet av <code class="varname">a</code> blir j,i-elementet av <strong
class="userinput"><code>a.'</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,c)</code></strong></span></dt><dd><p>Hämta element för en matris i rad <code
class=
"varname">b</code> och kolumn <code class="varname">c</code>. Om <code class="varname">b</code>, <code
class="varname">c</code> är vektorer så ger detta de motsvarande raderna, kolumnerna eller
delmatriserna.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,)</code></strong></span></dt><dd><p>Hämta rad av en matris (eller flera rader om
<code class="varname">b</code> är en vektor).</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b,:)</code></strong></span></dt><dd><p>Samma som ovan.</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(,c)</code></strong></span></dt><dd><p>Hämta kolumn av en
matris (eller flera kolumner om <code class="varname">b</code> är en vektor).</p></dd><dt><span
class="term"><strong class="userinput"><code>a@(:,c)</code></strong></span></dt><dd><p>Samma som
ovan.</p></dd><dt><span class="term"><strong
class="userinput"><code>a@(b)</code></strong></span></dt><dd><p>Hämta ett element från en matri
s behandlad som en vektor. Detta kommer traversera matrisen radvis.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b</code></strong></span></dt><dd><p>Bygg en vektor från <code
class="varname">a</code> till <code class="varname">b</code> (eller ange en rad- och kolumnregion för <code
class="literal">@</code>-operatorn). Till exempel kan vi för att få raderna 2 till 4 av matrisen <code
class="varname">A</code> göra </p><pre class="programlisting">A@(2:4,)
</pre><p> eftersom <strong class="userinput"><code>2:4</code></strong> kommer returnera en
vektor <strong class="userinput"><code>[2,3,4]</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>a:b:c</code></strong></span></dt><dd><p>Bygg en vektor från <code
class="varname">a</code> till <code class="varname">c</code> med <code class="varname">b</code> som
steglängd. Det vill säga exempelvis </p><pre class="programlisting">genius> 1:2:9
=
`[1, 3, 5, 7, 9]
-</pre><p>Då de inblandade talen är flyttal, till exempel <strong
class="userinput"><code>1.0:0.4:3.0</code></strong>, är utmatningen vad som förväntas även om att lägga till
0.4 till 1.0 fem gånger faktiskt är något mindre än 3.0 på grund av sättet som flyttal lagras i bas 2 (det
finns inget 0.4, det faktiska lagrade talet är bara något större). Sättet detta hanteras är detsamma som i
for-, sum-, och prod-slingorna. Om slutet är inom <strong class="userinput"><code>2^-20</code></strong>
gånger stegstorleken till ändpunkten, används ändpunkten och vi antar att det fanns avrundningsfel. Detta är
inte perfekt, men hanterar de flesta fallen. Denna kontroll görs bara från version 1.0.18 och framåt, så
exekvering av din kod kan skilja sig åt i äldre versioner. Använd faktiska rationella tal om du vill undvika
att hantera detta problem, möjligen tillsammans med <code class="function">float</code> om du vill få flyttal
i slutet. Till exempel gör <st
rong class="userinput"><code>1:2/5:3</code></strong> rätt sak och <strong
class="userinput"><code>float(1:2/5:3)</code></strong> ger dig till och med flyttal och är även något mer
exakt än <strong class="userinput"><code>1.0:0.4:3.0</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Skapa ett imaginärt tal (multiplicera <code
class="varname">a</code> med det imaginära). Observera att <code class="varname">i</code> vanligen skrivs
<strong class="userinput"><code>1i</code></strong>, så det ovanstående är detsamma som </p><pre
class="programlisting">(a)*1i
+</pre><p>Då de inblandade talen är flyttal, till exempel <strong
class="userinput"><code>1.0:0.4:3.0</code></strong>, är utmatningen vad som förväntas även om att lägga till
0.4 till 1.0 fem gånger faktiskt är något mindre än 3.0 på grund av sättet som flyttal lagras i bas 2 (det
finns inget 0.4, det faktiska lagrade talet är bara något större). Sättet detta hanteras är detsamma som i
for-, sum-, och prod-slingorna. Om slutet är inom <strong class="userinput"><code>2^-20</code></strong>
gånger stegstorleken till ändpunkten, används ändpunkten och vi antar att det fanns avrundningsfel. Detta är
inte perfekt, men hanterar de flesta fallen. Denna kontroll görs bara från version 1.0.18 och framåt, så
exekvering av din kod kan skilja sig åt i äldre versioner. Använd faktiska rationella tal om du vill undvika
att hantera detta problem, möjligen tillsammans med <code class="function">float</code> om du vill få flyttal
i slutet. Till exempel gör <st
rong class="userinput"><code>1:2/5:3</code></strong> rätt sak och <strong
class="userinput"><code>float(1:2/5:3)</code></strong> ger dig till och med flyttal och är även något mer
exakt än <strong class="userinput"><code>1.0:0.4:3.0</code></strong>.</p></dd><dt><span class="term"><strong
class="userinput"><code>(a)i</code></strong></span></dt><dd><p>Gör <code class="varname">a</code> till ett
imaginärt tal (multiplicera <code class="varname">a</code> med det imaginära). Observera att <code
class="varname">i</code> vanligen skrivs <strong class="userinput"><code>1i</code></strong>, så det
ovanstående är detsamma som </p><pre class="programlisting">(a)*1i
</pre></dd><dt><span class="term"><strong
class="userinput"><code>`a</code></strong></span></dt><dd><p>Citera en identifierare så att den inte
evalueras. Eller citera en matris så att den inte expanderas.</p></dd><dt><span class="term"><strong
class="userinput"><code>a swapwith b</code></strong></span></dt><dd><p>Byt värde på <code
class="varname">a</code> med värdet av <code class="varname">b</code>. Opererar för närvarande inte på
intervall av matriselement. Det returnerar <code class="constant">null</code>. Tillgängligt från version
1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment
a</code></strong></span></dt><dd><p>Inkrementera variabeln <code class="varname">a</code> med 1. Om <code
class="varname">a</code> är en matris inkrementeras varje element. Detta är ekvivalent med <strong
class="userinput"><code>a=a+1</code></strong>, men är något snabbare. Det returnerar <code
class="constant">null</code>. Tillgängligt från
version 1.0.13.</p></dd><dt><span class="term"><strong class="userinput"><code>increment a by
b</code></strong></span></dt><dd><p>Inkrementera variabeln <code class="varname">a</code> med <code
class="varname">b</code>. Om <code class="varname">a</code> är en matris inkrementeras varje element. Detta
är ekvivalent med <strong class="userinput"><code>a=a+b</code></strong>, men är något snabbare. Det
returnerar <code class="constant">null</code>. Tillgängligt från version 1.0.13.</p></dd></dl></div><div
class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Notera</h3><p>@()-operatorn
gör :-operatorn mest användbar. Med denna kan du ange regioner i en matris. Därmed är a@(2:4,6) raderna 2,3,4
för kolumn 6. Eller så ger a@(,1:2) dig de två första kolumnerna i en matris. Du kan också tilldela till
@()-operatorn, så länge som högervärdet är en matris som matchar regionens storlek, eller om det är någon
annan sorts värde.</p></div><d
iv class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Notera</h3><p>Jämförelseoperatorerna (förutom <=>-operatorn, vilken beter sig normalt) är
inte strikt binära operatorer, de kan i själva verket grupperas på det vanliga matematiska sättet, t.ex. så
är (1<x<=y<5) ett giltigt booleskt uttryck och betyder precis vad det borde, det vill säga (1<x
och x≤y och y<5)</p></div><div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"><h3
class="title">Notera</h3><p>Unärt minus opererar annorlunda beroende på var det förekommer. Om det förekommer
före ett tal binder det väldigt hårt, om det förekommer före ett uttryck binder det mindre hårt än potens-
och fakultet-operatorerna. Så till exempel är <strong class="userinput"><code>-1^k</code></strong> faktiskt
<strong class="userinput"><code>(-1)^k</code></strong>, men <strong
class="userinput"><code>-foo(1)^k</code></strong> är verkligen <strong cla
ss="userinput"><code>-(foo(1)^k)</code></strong>. Så var aktsam över hur du använder det, och om du är
osäker, lägg till parenteser.</p></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch05s06.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch05.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch06.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Moduloberäkning </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%" align="right"
valign="top"> Kapitel 6. Programmering med GEL</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch06s05.html b/help/sv/html/ch06s05.html
index 9b3ecae..8d08704 100644
--- a/help/sv/html/ch06s05.html
+++ b/help/sv/html/ch06s05.html
@@ -1,4 +1,4 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Globala variabler och
räckvidd för variabler</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch06.html" title="Kapitel 6.
Programmering med GEL"><link rel="prev" href="ch06s04.html" title="Jämförelseoperatorer"><link rel="next"
href="ch06s06.html" title="Parametervariabler"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Globala variabler och räckvidd för variabler</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Föregående</a> </td><th width="60%"
align="center">Kapitel 6. Programmering med GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Nästa</a></td></tr></table><hr></d
iv><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Globala variabler och räckvidd för
variabler</h2></div></div></div><p>GEL är ett <a class="ulink"
href="http://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top"> språk med dynamisk räckvidd</a>.
Vi kommer att förklara vad detta betyder nedan. Det betyder att normala variabler och funktioner har dynamisk
räckvidd. Undantaget är <a class="link" href="ch06s06.html" title="Parametervariabler">parametervariabler</a>
som alltid är globala.</p><p>Som de flesta programmeringsspråk har GEL olika typer av variabler. Normalt då
en variabel är definierad i en funktion är den synlig från den funktionen och från alla funktioner som
anropas (alla högre kontexter). Till exempel, anta att en funktion <code class="function">f</code> definierar
en variabel <code class="varname">a</code> och sedan anropar funktionen <code class="functi
on">g</code>. Då kan funktion <code class="function">g</code> referera till <code class="varname">a</code>.
Men då <code class="function">f</code> returnerar, går variabeln <code class="varname">a</code> utom
räckvidd. Till exempel kommer den följande koden att skriva ut 5. Funktionen <code class="function">g</code>
kan inte anropas på toppnivån (utanför <code class="function">f</code> eftersom <code
class="varname">a</code> inte kommer vara definierad). </p><pre class="programlisting">function f() = (a:=5;
g());
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Globala variabler och
räckvidd för variabler</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link
rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch06.html" title="Kapitel 6.
Programmering med GEL"><link rel="prev" href="ch06s04.html" title="Jämförelseoperatorer"><link rel="next"
href="ch06s06.html" title="Parametervariabler"></head><body bgcolor="white" text="black" link="#0000FF"
vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Globala variabler och räckvidd för variabler</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch06s04.html">Föregående</a> </td><th width="60%"
align="center">Kapitel 6. Programmering med GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch06s06.html">Nästa</a></td></tr></table><hr></d
iv><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-variables-global"></a>Globala variabler och räckvidd för
variabler</h2></div></div></div><p>GEL är ett <a class="ulink"
href="https://en.wikipedia.org/wiki/Scope_%28programming%29"; target="_top"> språk med dynamisk räckvidd</a>.
Vi kommer att förklara vad detta betyder nedan. Det betyder att normala variabler och funktioner har dynamisk
räckvidd. Undantaget är <a class="link" href="ch06s06.html" title="Parametervariabler">parametervariabler</a>
som alltid är globala.</p><p>Som de flesta programmeringsspråk har GEL olika typer av variabler. Normalt då
en variabel är definierad i en funktion är den synlig från den funktionen och från alla funktioner som
anropas (alla högre kontexter). Till exempel, anta att en funktion <code class="function">f</code> definierar
en variabel <code class="varname">a</code> och sedan anropar funktionen <code class="funct
ion">g</code>. Då kan funktion <code class="function">g</code> referera till <code class="varname">a</code>.
Men då <code class="function">f</code> returnerar, går variabeln <code class="varname">a</code> utom
räckvidd. Till exempel kommer den följande koden att skriva ut 5. Funktionen <code class="function">g</code>
kan inte anropas på toppnivån (utanför <code class="function">f</code> eftersom <code
class="varname">a</code> inte kommer vara definierad). </p><pre class="programlisting">function f() = (a:=5;
g());
function g() = print(a);
f();
</pre><p>Om du definierar en variabel inuti en funktion kommer den åsidosätta variabler definierade i
anropande funktioner. Som ett exempel modifierar vi koden ovan och skriver: </p><pre
class="programlisting">function f() = (a:=5; g());
diff --git a/help/sv/html/ch07s02.html b/help/sv/html/ch07s02.html
index 1cb211d..8860f1c 100644
--- a/help/sv/html/ch07s02.html
+++ b/help/sv/html/ch07s02.html
@@ -1,4 +1,14 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Toppnivåsyntax</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch07.html"
title="Kapitel 7. Avancerad programmering med GEL"><link rel="prev" href="ch07.html" title="Kapitel 7.
Avancerad programmering med GEL"><link rel="next" href="ch07s03.html" title="Returnera
funktioner"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Toppnivåsyntax</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch07.html">Föregående</a> </td><th width="60%" align="center">Kapitel 7. Avancerad programmering med
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch07s03.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-toplevel-syntax"></a>Toppnivåsyntax</h2></div></div></div><p>Syntaxen skiljer sig något
beroende på om du matar in satser på toppnivån gentemot då de används inom parenteser eller i funktioner. På
toppnivån uppför sig retur just som om du tryckte retur på kommandoraden. Tänk därför på program som bara en
följd av rader som matats in på kommandoraden. I synnerhet behöver du inte ange avskiljaren i slutet på raden
(om den förstås inte är del av flera satser inom parenteser).</p><p>Följande kod producerar ett fel då den
matas in i toppnivån för ett program, medan den kommer fungera fint i en funktion. </p><pre
class="programlisting">if Ngt() then
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Toppnivåsyntax</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch07.html"
title="Kapitel 7. Avancerad programmering med GEL"><link rel="prev" href="ch07.html" title="Kapitel 7.
Avancerad programmering med GEL"><link rel="next" href="ch07s03.html" title="Returnera
funktioner"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Toppnivåsyntax</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch07.html">Föregående</a> </td><th width="60%" align="center">Kapitel 7. Avancerad programmering med
GEL</th><td width="20%" align="right"> <a accesskey="n"
href="ch07s03.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear: both"><a
name="genius-gel-toplevel-syntax"></a>Toppnivåsyntax</h2></div></div></div><p>Syntaxen skiljer sig något
beroende på om du matar in satser på toppnivån gentemot då de används inom parenteser eller i funktioner. På
toppnivån uppför sig retur just som om du tryckte retur på kommandoraden. Tänk därför på program som bara en
följd av rader som matats in på kommandoraden. I synnerhet behöver du inte ange avskiljaren i slutet på raden
(om den förstås inte är del av flera satser inom parenteser). Då en sats inte avslutas med en avskiljare på
toppnivån skrivs resultatet ut efter körning.</p><p>Till exempel kommer </p><pre
class="programlisting">function f(x)=x^2
+f(3)
+</pre><p> att först skriva ut resultatet av att ställa in en funktion (en representation av funktionen, i
det här fallet <code class="computeroutput">(`(x)=(x^2))</code>) och sedan det förväntade 9. För att undvika
detta, ange en avskiljare efter funktionsdefinitionen. </p><pre class="programlisting">function f(x)=x^2;
+f(3)
+</pre><p> Om du behöver stoppa en avskiljare i din funktion måste den omges av parenteser. Till exempel:
</p><pre class="programlisting">function f(x)=(
+ y=1;
+ for j=1 to x do
+ y = y+j;
+ y^2
+);
+</pre><p>Följande kod producerar ett fel då den matas in i toppnivån för ett program, medan den kommer
fungera fint i en funktion. </p><pre class="programlisting">if Ngt() then
ExekveraNgt()
else
ExekveraNgtAnnat()
diff --git a/help/sv/html/ch11s04.html b/help/sv/html/ch11s04.html
index 35101ee..0cd8cec 100644
--- a/help/sv/html/ch11s04.html
+++ b/help/sv/html/ch11s04.html
@@ -1 +1 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Konstanter</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s03.html" title="Parametrar"><link
rel="next" href="ch11s05.html" title="Numeriska funktioner"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Konstanter</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s03.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="titl
e" style="clear: both"><a
name="genius-gel-function-list-constants"></a>Konstanter</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Catalans konstant, ungefär 0.915... Den är definierad som serien där
termerna är <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, där <code
class="varname">k</code> går från 0 till oändligheten.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Catalan%27s_constant"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/CatalansConstant.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alias: <code class="function">gamma</code></p><p>Eulers gammakonstant.
Ibland kal
lad Euler-Mascheroni-konstanten.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html"; target="_top">Mathworld</a>
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>Det
gyllene snittet.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Golden_ratio";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/GoldenRatio";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre class="synopsis">Gravity<
/pre><p>Acceleration vid fritt fall vid havsytan i meter per sekundkvadrat- Detta är den vanliga
gravitationskonstanten 9.80665. Gravitationen i dina hemtrakter kan skilja sig från denna på grund av annan
höjd och för att jorden inte är ett perfekt klot.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>Basen för den naturliga logaritmen. <strong
class="userinput"><code>e^x</code></strong> är den exponentiella funktionen <a class="link"
href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a>. Den är ungefär 2.71828182846...
Detta tal kallas ibland Eulers tal, men det finns flera tal som också kallas Eulers. Ett exempel på det är
gammakonstanten: <a class="link" href="ch11s04.html#gel-function-EulerConstant"><code
class="function">EulerConstant</code></a>.</p>
<p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/E";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/e.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-pi"></a>pi</span></dt><dd><pre class="synopsis">pi</pre><p>Talet pi, det vill säga
förhållandet mellan en cirkels omkrets och dess diameter. Detta är ungefär 3,14159265359...</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/Pi"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s03.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Parametrar </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%" align="right"
valign="top"> Numeriska funktioner</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Konstanter</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s03.html" title="Parametrar"><link
rel="next" href="ch11s05.html" title="Numeriska funktioner"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Konstanter</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s03.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s05.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="titl
e" style="clear: both"><a
name="genius-gel-function-list-constants"></a>Konstanter</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-CatalanConstant"></a>CatalanConstant</span></dt><dd><pre
class="synopsis">CatalanConstant</pre><p>Catalans konstant, ungefär 0.915... Den är definierad som serien där
termerna är <strong class="userinput"><code>(-1^k)/((2*k+1)^2)</code></strong>, där <code
class="varname">k</code> går från 0 till oändligheten.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Catalan%27s_constant"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/CatalansConstant.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-EulerConstant"></a>EulerConstant</span></dt><dd><pre
class="synopsis">EulerConstant</pre><p>Alias: <code class="function">gamma</code></p><p>Eulers gammakonstant.
Ibland ka
llad Euler-Mascheroni-konstanten.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Euler-Mascheroni_constant"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/MascheroniConstant"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html"; target="_top">Mathworld</a>
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-GoldenRatio"></a>GoldenRatio</span></dt><dd><pre class="synopsis">GoldenRatio</pre><p>Det
gyllene snittet.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Golden_ratio";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/GoldenRatio";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/GoldenRatio.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Gravity"></a>Gravity</span></dt><dd><pre class="synopsis">Gravi
ty</pre><p>Acceleration vid fritt fall vid havsytan i meter per sekundkvadrat- Detta är den vanliga
gravitationskonstanten 9.80665. Gravitationen i dina hemtrakter kan skilja sig från denna på grund av annan
höjd och för att jorden inte är ett perfekt klot.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Standard_gravity"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-e"></a>e</span></dt><dd><pre
class="synopsis">e</pre><p>Basen för den naturliga logaritmen. <strong
class="userinput"><code>e^x</code></strong> är den exponentiella funktionen <a class="link"
href="ch11s05.html#gel-function-exp"><code class="function">exp</code></a>. Den är ungefär 2.71828182846...
Detta tal kallas ibland Eulers tal, men det finns flera tal som också kallas Eulers. Ett exempel på det är
gammakonstanten: <a class="link" href="ch11s04.html#gel-function-EulerConstant"><code
class="function">EulerConstant</code></a>.
</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)"
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/E";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/e.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-pi"></a>pi</span></dt><dd><pre class="synopsis">pi</pre><p>Talet pi, det vill säga
förhållandet mellan en cirkels omkrets och dess diameter. Detta är ungefär 3,14159265359...</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Pi"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/Pi"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/Pi.html"; target="_top">Mathworld</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesske
y="p" href="ch11s03.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s05.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Parametrar </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%" align="right"
valign="top"> Numeriska funktioner</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s05.html b/help/sv/html/ch11s05.html
index 5a6a607..5f85b73 100644
--- a/help/sv/html/ch11s05.html
+++ b/help/sv/html/ch11s05.html
@@ -1,8 +1,8 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Numeriska
funktioner</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över
GEL-funktioner"><link rel="prev" href="ch11s04.html" title="Konstanter"><link rel="next" href="ch11s06.html"
title="Trigonometri"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Numeriska funktioner</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s04.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-function-list-numeric"></a>Numeriska
funktioner</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Alias: <code class="function">abs</code></p><p>Absolutbeloppet av ett tal, och om <code
class="varname">x</code> är ett komplext tal så är detta avståndet för <code class="varname">x</code> till
origo. Detta är ekvivalent med <strong class="userinput"><code>|x|</code></strong>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolutbelopp)</a>, <a class="ulink"
href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath (modulus)</a>, <a class="ulink"
href="http://mathworld.wolfram.com/AbsoluteValue.
html" target="_top">Mathworld (absolutbelopp)</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld (complex modulus)</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Ersätt väldigt litet tal med noll.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alias: <code class="function">conj</code><code
class="function">Conj</code></p><p>Beräknar komplexkonjugatet av det komplexa talet <code
class="varname">z</code>. Om <code class="varname">z</code> är en vektor eller matris konjugeras alla dess
element.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span></
dt><dd><pre class="synopsis">Denominator (x)</pre><p>Hämta nämnaren för ett rationellt tal.</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Returnera bråkdelen av ett tal.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alias: <code class="function">ImaginaryPart</code></p><p>Hämta den imaginära
delen av ett komplext tal. Till exempel ger <strong class="userinput"><code>Re(3+4i)</code></strong> svaret
4.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
för mer information.</p></dd><dt><span class=
"term"><a name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre
class="synopsis">IntegerQuotient (m,n)</pre><p>Division utan rest.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(tal)</pre><p>Kontrollera om argumentet är ett komplext (icke-reellt) tal. Observera att vi menar icke-reellt
tal. Det vill säga <strong class="userinput"><code>IsComplex(3)</code></strong> ger false, medan <strong
class="userinput"><code>IsComplex(3-1i)</code></strong> ger true.</p></dd><dt><span class="term"><a
name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (tal)</pre><p>Kontrollera om argumentet är ett möjligtvis komplext
rationellt tal. Det vill säga om både real- och imaginärdelarna anges som rationella tal. Givetvis betyder
rationell helt enkelt ”inte lagrad som ett flyttal”.</p></dd><dt><span class="term"><a name="gel
-function-IsFloat"></a>IsFloat</span></dt><dd><pre class="synopsis">IsFloat (tal)</pre><p>Kontrollera om
argumentet är ett reellt flyttal (icke-komplext).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(tal)</pre><p>Alias: <code class="function">IsComplexInteger</code></p><p>Kontrollera om argumentet är ett
möjligtvis komplext heltal. Det vill säga ett komplext heltal är ett heltal på formen <strong
class="userinput"><code>n+1i*m</code></strong> där <code class="varname">n</code> och <code
class="varname">m</code> är heltal.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(tal)</pre><p>Kontrollera om argumentet är ett heltal (icke-komplext).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteger (t
al)</pre><p>Kontrollera om argumentet är ett icke-negativt reellt heltal. Det vill säga antingen ett
positivt heltal eller noll.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (tal)</pre><p>Alias: <code
class="function">IsNaturalNumber</code></p><p>Kontrollera om argumentet är ett positivt reellt heltal.
Observera att vi accepterar konventionen att 0 inte är ett naturligt tal.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(tal)</pre><p>Kontrollera om argumentet är ett rationellt tal (icke-komplext). Rationellt betyder förstås
endast ”inte lagrat som ett flyttal”.</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (tal)</pre><p>Kontrollera
om argumentet är ett reellt tal.</p></dd><dt><span class="term"><a name="gel-func
tion-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Hämta täljaren för
ett rationellt tal.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Numerator";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Re"></a>Re</span></dt><dd><pre class="synopsis">Re (z)</pre><p>Alias: <code
class="function">RealPart</code></p><p>Hämta den reella delen av ett komplext tal. Till exempel ger <strong
class="userinput"><code>Re(3+4i)</code></strong> svaret 3.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Alias: <code class="function">sign</code></p><p>Returnera tecknet för ett
tal. Det vill säga returnerar <code class="literal">-1</code> om värdet är negativt, <code
class="literal">0</code> o
m värdet är noll och <code class="literal">1</code> om värdet är positivt. Om <code class="varname">x</code>
är ett komplext värde så returnerar <code class="function">Sign</code> riktningen eller 0.</p></dd><dt><span
class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre class="synopsis">ceil
(x)</pre><p>Alias: <code class="function">Ceiling</code></p><p>Hämta det minsta heltalet större än eller lika
med <code class="varname">n</code>. Exempel: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>ceil(1.1)</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Numeriska
funktioner</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home"
href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över
GEL-funktioner"><link rel="prev" href="ch11s04.html" title="Konstanter"><link rel="next" href="ch11s06.html"
title="Trigonometri"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Numeriska funktioner</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s04.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s06.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2
class="title" style="clear: both"><a name="genius-gel-function-list-numeric"></a>Numeriska
funktioner</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AbsoluteValue"></a>AbsoluteValue</span></dt><dd><pre class="synopsis">AbsoluteValue
(x)</pre><p>Alias: <code class="function">abs</code></p><p>Absolutbeloppet av ett tal, och om <code
class="varname">x</code> är ett komplext tal så är detta avståndet för <code class="varname">x</code> till
origo. Detta är ekvivalent med <strong class="userinput"><code>|x|</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Absolute_value"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/AbsoluteValue"; target="_top">Planetmath (absolutbelopp)</a>, <a class="ulink"
href="http://planetmath.org/ModulusOfComplexNumber"; target="_top">Planetmath (modulus)</a>, <a class="ulink"
href="http://mathworld.wolfram.com/AbsoluteValue
.html" target="_top">Mathworld (absolutbelopp)</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/ComplexModulus.html"; target="_top">Mathworld (komplex modulus)</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Chop"></a>Chop</span></dt><dd><pre
class="synopsis">Chop (x)</pre><p>Ersätt väldigt litet tal med noll.</p></dd><dt><span class="term"><a
name="gel-function-ComplexConjugate"></a>ComplexConjugate</span></dt><dd><pre
class="synopsis">ComplexConjugate (z)</pre><p>Alias: <code class="function">conj</code><code
class="function">Conj</code></p><p>Beräknar komplexkonjugatet av det komplexa talet <code
class="varname">z</code>. Om <code class="varname">z</code> är en vektor eller matris konjugeras alla dess
element.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Complex_conjugate";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Denominator"></a>Denominator</span>
</dt><dd><pre class="synopsis">Denominator (x)</pre><p>Hämta nämnaren för ett rationellt tal.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Denominator"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-FractionalPart"></a>FractionalPart</span></dt><dd><pre class="synopsis">FractionalPart
(x)</pre><p>Returnera bråkdelen av ett tal.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fractional_part"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Im"></a>Im</span></dt><dd><pre
class="synopsis">Im (z)</pre><p>Alias: <code class="function">ImaginaryPart</code></p><p>Hämta den imaginära
delen av ett komplext tal. Till exempel ger <strong class="userinput"><code>Re(3+4i)</code></strong> svaret
4.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Imaginary_part"; target="_top">Wikipedia</a>
för mer information.</p></dd><dt><span c
lass="term"><a name="gel-function-IntegerQuotient"></a>IntegerQuotient</span></dt><dd><pre
class="synopsis">IntegerQuotient (m,n)</pre><p>Division utan rest.</p></dd><dt><span class="term"><a
name="gel-function-IsComplex"></a>IsComplex</span></dt><dd><pre class="synopsis">IsComplex
(tal)</pre><p>Kontrollera om argumentet är ett komplext (icke-reellt) tal. Observera att vi menar icke-reellt
tal. Det vill säga <strong class="userinput"><code>IsComplex(3)</code></strong> ger false, medan <strong
class="userinput"><code>IsComplex(3-1i)</code></strong> ger true.</p></dd><dt><span class="term"><a
name="gel-function-IsComplexRational"></a>IsComplexRational</span></dt><dd><pre
class="synopsis">IsComplexRational (tal)</pre><p>Kontrollera om argumentet är ett möjligtvis komplext
rationellt tal. Det vill säga om både real- och imaginärdelarna anges som rationella tal. Givetvis betyder
rationell helt enkelt ”inte lagrad som ett flyttal”.</p></dd><dt><span class="term"><a name
="gel-function-IsFloat"></a>IsFloat</span></dt><dd><pre class="synopsis">IsFloat (tal)</pre><p>Kontrollera
om argumentet är ett reellt flyttal (icke-komplext).</p></dd><dt><span class="term"><a
name="gel-function-IsGaussInteger"></a>IsGaussInteger</span></dt><dd><pre class="synopsis">IsGaussInteger
(tal)</pre><p>Alias: <code class="function">IsComplexInteger</code></p><p>Kontrollera om argumentet är ett
möjligtvis komplext heltal. Det vill säga ett komplext heltal är ett heltal på formen <strong
class="userinput"><code>n+1i*m</code></strong> där <code class="varname">n</code> och <code
class="varname">m</code> är heltal.</p></dd><dt><span class="term"><a
name="gel-function-IsInteger"></a>IsInteger</span></dt><dd><pre class="synopsis">IsInteger
(tal)</pre><p>Kontrollera om argumentet är ett heltal (icke-komplext).</p></dd><dt><span class="term"><a
name="gel-function-IsNonNegativeInteger"></a>IsNonNegativeInteger</span></dt><dd><pre
class="synopsis">IsNonNegativeInteg
er (tal)</pre><p>Kontrollera om argumentet är ett icke-negativt reellt heltal. Det vill säga antingen ett
positivt heltal eller noll.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveInteger"></a>IsPositiveInteger</span></dt><dd><pre
class="synopsis">IsPositiveInteger (tal)</pre><p>Alias: <code
class="function">IsNaturalNumber</code></p><p>Kontrollera om argumentet är ett positivt reellt heltal.
Observera att vi accepterar konventionen att 0 inte är ett naturligt tal.</p></dd><dt><span class="term"><a
name="gel-function-IsRational"></a>IsRational</span></dt><dd><pre class="synopsis">IsRational
(tal)</pre><p>Kontrollera om argumentet är ett rationellt tal (icke-komplext). Rationellt betyder förstås
endast ”inte lagrat som ett flyttal”.</p></dd><dt><span class="term"><a
name="gel-function-IsReal"></a>IsReal</span></dt><dd><pre class="synopsis">IsReal (tal)</pre><p>Kontrollera
om argumentet är ett reellt tal.</p></dd><dt><span class="term"><a name="gel
-function-Numerator"></a>Numerator</span></dt><dd><pre class="synopsis">Numerator (x)</pre><p>Hämta täljaren
för ett rationellt tal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Numerator";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Re"></a>Re</span></dt><dd><pre class="synopsis">Re (z)</pre><p>Alias: <code
class="function">RealPart</code></p><p>Hämta den reella delen av ett komplext tal. Till exempel ger <strong
class="userinput"><code>Re(3+4i)</code></strong> svaret 3.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Real_part"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Sign"></a>Sign</span></dt><dd><pre
class="synopsis">Sign (x)</pre><p>Alias: <code class="function">sign</code></p><p>Returnera tecknet för ett
tal. Det vill säga returnerar <code class="literal">-1</code> om värdet är negativt, <code class="literal">0</
code> om värdet är noll och <code class="literal">1</code> om värdet är positivt. Om <code
class="varname">x</code> är ett komplext värde så returnerar <code class="function">Sign</code> riktningen
eller 0.</p></dd><dt><span class="term"><a name="gel-function-ceil"></a>ceil</span></dt><dd><pre
class="synopsis">ceil (x)</pre><p>Alias: <code class="function">Ceiling</code></p><p>Hämta det minsta
heltalet större än eller lika med <code class="varname">n</code>. Exempel: </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>ceil(1.1)</code></strong>
= 2
<code class="prompt">genius></code> <strong class="userinput"><code>ceil(-1.1)</code></strong>
= -1
-</pre><p>Observera att du bör vara försiktig och notera att flyttal lagras binärt och därför kanske inte är
vad du förväntar dig. Till exempel har vi <strong class="userinput"><code>ceil(420/4.2)</code></strong> som
returnerar 101 istället för det förväntade 100. Detta är för att 4.2 faktiskt är något mindre än 4.2. Använd
bråkrepresentationen <strong class="userinput"><code>42/10</code></strong> om du vill ha exakt
aritmetik.</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>Exponentialfunktionen. Detta är funktionen <strong
class="userinput"><code>e^x</code></strong> där <code class="varname">e</code> är <a class="link"
href="ch11s04.html#gel-function-e">basen för den naturliga logaritmen</a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Exponential_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/LogarithmFunction"; target="_to
p">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-float"></a>float</span></dt><dd><pre class="synopsis">float (x)</pre><p>Gör ett tal till
ett flyttalsvärde. Det vill säga returnerar flyttalsrepresentationen av talet <code
class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Alias: <code
class="function">Floor</code></p><p>Hämta det största heltalet mindre än eller lika med <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>Den naturliga logaritmen, logaritmen med bas <code
class="varname">e</code>.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> eller <a class="ulin
k" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/NaturalLogarithm.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logaritm för <code
class="varname">x</code> med basen <code class="varname">b</code> (anropar <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> om i moduloläge),
om bas inte är angiven används <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logaritmen av
<code class="varname">x</code> bas 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsis"
log2 (x)</pre><p>Alias: <code class="function">lg</code></p><p>Logaritmen av <code class="varname">x</code>
bas 2.</p></dd><dt><span class="term"><a name="gel-function-max"></a>max</span></dt><dd><pre
class="synopsis">max (a,arg...)</pre><p>Alias: <code class="function">Max</code><code
class="function">Maximum</code></p><p>Returnera maximum av argument eller matris.</p></dd><dt><span
class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre class="synopsis">min
(a,arg...)</pre><p>Alias: <code class="function">Min</code><code
class="function">Minimum</code></p><p>Returnera minimum av argument eller matris.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(storlek...)</pre><p>Generera slumpmässigt flyttal i intervallet <code class="literal">[0,1)</code>. Om
storlek är angiven returneras en matris (om två tal anges) eller en vektor (om ett tal anges) av den
angivna storleken.</p></dd><dt><span class="ter
m"><a name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,storlek...)</pre><p>Generera slumpmässigt heltal i intervallet <code class="literal">[0,1)</code>. Om
storlek är angiven returneras en matris (om två tal anges) eller en vektor (om ett tal anges) av den angivna
storleken. Till exempel, </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
+</pre><p>Observera att du bör vara försiktig och notera att flyttal lagras binärt och därför kanske inte är
vad du förväntar dig. Till exempel har vi <strong class="userinput"><code>ceil(420/4.2)</code></strong> som
returnerar 101 istället för det förväntade 100. Detta är för att 4.2 faktiskt är något mindre än 4.2. Använd
bråkrepresentationen <strong class="userinput"><code>42/10</code></strong> om du vill ha exakt
aritmetik.</p></dd><dt><span class="term"><a name="gel-function-exp"></a>exp</span></dt><dd><pre
class="synopsis">exp (x)</pre><p>Exponentialfunktionen. Detta är funktionen <strong
class="userinput"><code>e^x</code></strong> där <code class="varname">e</code> är <a class="link"
href="ch11s04.html#gel-function-e">basen för den naturliga logaritmen</a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Exponential_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/LogarithmFunction"; target="_t
op">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/ExponentialFunction.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-float"></a>float</span></dt><dd><pre class="synopsis">float (x)</pre><p>Gör ett tal till
ett flyttalsvärde. Det vill säga returnerar flyttalsrepresentationen av talet <code
class="varname">x</code>.</p></dd><dt><span class="term"><a
name="gel-function-floor"></a>floor</span></dt><dd><pre class="synopsis">floor (x)</pre><p>Alias: <code
class="function">Floor</code></p><p>Hämta det största heltalet mindre än eller lika med <code
class="varname">n</code>.</p></dd><dt><span class="term"><a name="gel-function-ln"></a>ln</span></dt><dd><pre
class="synopsis">ln (x)</pre><p>Den naturliga logaritmen, logaritmen med bas <code
class="varname">e</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Natural_logarithm";
target="_top">Wikipedia</a> eller <a class="ul
ink" href="http://planetmath.org/LogarithmFunction"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/NaturalLogarithm.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-log"></a>log</span></dt><dd><pre
class="synopsis">log (x)</pre><pre class="synopsis">log (x,b)</pre><p>Logaritm för <code
class="varname">x</code> med basen <code class="varname">b</code> (anropar <a class="link"
href="ch11s07.html#gel-function-DiscreteLog"><code class="function">DiscreteLog</code></a> om i moduloläge),
om bas inte är angiven används <a class="link" href="ch11s04.html#gel-function-e"><code
class="varname">e</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-log10"></a>log10</span></dt><dd><pre class="synopsis">log10 (x)</pre><p>Logaritmen av
<code class="varname">x</code> bas 10.</p></dd><dt><span class="term"><a
name="gel-function-log2"></a>log2</span></dt><dd><pre class="synopsi
s">log2 (x)</pre><p>Alias: <code class="function">lg</code></p><p>Logaritmen av <code
class="varname">x</code> bas 2.</p></dd><dt><span class="term"><a
name="gel-function-max"></a>max</span></dt><dd><pre class="synopsis">max (a,arg...)</pre><p>Alias: <code
class="function">Max</code><code class="function">Maximum</code></p><p>Returnera maximum av argument eller
matris.</p></dd><dt><span class="term"><a name="gel-function-min"></a>min</span></dt><dd><pre
class="synopsis">min (a,arg...)</pre><p>Alias: <code class="function">Min</code><code
class="function">Minimum</code></p><p>Returnera minimum av argument eller matris.</p></dd><dt><span
class="term"><a name="gel-function-rand"></a>rand</span></dt><dd><pre class="synopsis">rand
(storlek...)</pre><p>Generera slumpmässigt flyttal i intervallet <code class="literal">[0,1)</code>. Om
storlek är angiven returneras en matris (om två tal anges) eller en vektor (om ett tal anges) av den angivna
storleken.</p></dd><dt><span class="t
erm"><a name="gel-function-randint"></a>randint</span></dt><dd><pre class="synopsis">randint
(max,storlek...)</pre><p>Generera slumpmässigt heltal i intervallet <code class="literal">[0,1)</code>. Om
storlek är angiven returneras en matris (om två tal anges) eller en vektor (om ett tal anges) av den angivna
storleken. Till exempel, </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>randint(4)</code></strong>
= 3
<code class="prompt">genius></code> <strong class="userinput"><code>randint(4,2)</code></strong>
=
diff --git a/help/sv/html/ch11s06.html b/help/sv/html/ch11s06.html
index c0575b4..36cac45 100644
--- a/help/sv/html/ch11s06.html
+++ b/help/sv/html/ch11s06.html
@@ -1,2 +1,2 @@
-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometri</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s05.html" title="Numeriska
funktioner"><link rel="next" href="ch11s07.html" title="Talteori"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometri</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="ti
tle" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonometri</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alias: <code
class="function">arccos</code></p><p>arccos-funktionen (invers cos).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Alias: <code
class="function">arccosh</code></p><p>arccosh-funktionen (invers cosh).</p></dd><dt><span class="term"><a
name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Alias: <code
class="function">arccot</code></p><p>arccot-funktionen (invers cot).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Alias: <code
class="function">arccoth</code></p><p>arccoth-funktionen (invers coth).</p></d
d><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc
(x)</pre><p>Alias: <code class="function">arccsc</code></p><p>Inversa cosekantfunktionen.</p></dd><dt><span
class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch
(x)</pre><p>Alias: <code class="function">arccsch</code></p><p>Inversa hyperboliska
cosekantfunktionen.</p></dd><dt><span class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre
class="synopsis">asec (x)</pre><p>Alias: <code class="function">arcsec</code></p><p>Inversa
sekantfunktionen.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Alias: <code class="function">arcsech</code></p><p>Inversa hyperboliska
sekantfunktionen.</p></dd><dt><span class="term"><a name="gel-function-asin"></a>asin</span></dt><dd><pre
class="synopsis">asin (x)</pre><p>Alias: <code class="function">arcsin</code>
</p><p>arcsin-funktionen (invers sin).</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Alias: <code
class="function">arcsinh</code></p><p>arcsinh-funktionen (invers sinh).</p></dd><dt><span class="term"><a
name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Alias: <code
class="function">arctan</code></p><p>Beräknar arcustangensfunktionen (invers tangens).</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alias: <code class="function">arctanh</code></p><p>arctanh-funktionen
(invers tanh).</p></dd><dt><span class="term"><a name="gel-function-atan2"></a>atan2</span><
/dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alias: <code
class="function">arctan2</code></p><p>Beräknar arctan2-funktionen. Om <strong
class="userinput"><code>x>0</code></strong> returnerar den <strong
class="userinput"><code>atan(y/x)</code></strong>. If <strong class="userinput"><code>x<0</code></strong>
returnerar den <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>. Då <strong
class="userinput"><code>x=0</code></strong> returnerar den <strong class="userinput"><code>sign(y) *
- pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> returnerar 0
snarare än att misslyckas.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Atan2";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cos"></a>cos</span></dt><dd><pre class="synopsis">cos (x)</pre><p>Beräknar
cosinusfunktionen.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cosh"></a>cosh</span></dt><dd><pre class="synopsis">cosh (x)</pre><p>Beräknar funktionen
för hyperbolisk cosinus.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_functi
on" target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cot"></a>cot</span></dt><dd><pre class="synopsis">cot
(x)</pre><p>Cotangensfunktionen.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Hyperboliska cotangensfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-csc"></a
csc</span></dt><dd><pre class="synopsis">csc (x)</pre><p>Cosekantfunktionen.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Hyperboliska cosekantfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Sekantfunktionen.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href=
"http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Hyperboliska sekantfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Beräknar sinusfunktionen.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">sinh (x
)</pre><p>Beräknar funktionen för hyperbolisk sinus.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Beräknar tangensfunktionen.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Hyperboliska tangensfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetma
th.org/HyperbolicFunctions" target="_top">Planetmath</a> för mer information.</p></dd></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s05.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s07.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Numeriska funktioner
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align="right" valign="top"> Talteori</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Trigonometri</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s05.html" title="Numeriska
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link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Trigonometri</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s05.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s07.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="ti
tle" style="clear: both"><a
name="genius-gel-function-list-trigonometry"></a>Trigonometri</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-acos"></a>acos</span></dt><dd><pre class="synopsis">acos (x)</pre><p>Alias: <code
class="function">arccos</code></p><p>arccos-funktionen (invers cos).</p></dd><dt><span class="term"><a
name="gel-function-acosh"></a>acosh</span></dt><dd><pre class="synopsis">acosh (x)</pre><p>Alias: <code
class="function">arccosh</code></p><p>arccosh-funktionen (invers cosh).</p></dd><dt><span class="term"><a
name="gel-function-acot"></a>acot</span></dt><dd><pre class="synopsis">acot (x)</pre><p>Alias: <code
class="function">arccot</code></p><p>arccot-funktionen (invers cot).</p></dd><dt><span class="term"><a
name="gel-function-acoth"></a>acoth</span></dt><dd><pre class="synopsis">acoth (x)</pre><p>Alias: <code
class="function">arccoth</code></p><p>arccoth-funktionen (invers coth).</p></d
d><dt><span class="term"><a name="gel-function-acsc"></a>acsc</span></dt><dd><pre class="synopsis">acsc
(x)</pre><p>Alias: <code class="function">arccsc</code></p><p>Inversa cosekantfunktionen.</p></dd><dt><span
class="term"><a name="gel-function-acsch"></a>acsch</span></dt><dd><pre class="synopsis">acsch
(x)</pre><p>Alias: <code class="function">arccsch</code></p><p>Inversa hyperboliska
cosekantfunktionen.</p></dd><dt><span class="term"><a name="gel-function-asec"></a>asec</span></dt><dd><pre
class="synopsis">asec (x)</pre><p>Alias: <code class="function">arcsec</code></p><p>Inversa
sekantfunktionen.</p></dd><dt><span class="term"><a name="gel-function-asech"></a>asech</span></dt><dd><pre
class="synopsis">asech (x)</pre><p>Alias: <code class="function">arcsech</code></p><p>Inversa hyperboliska
sekantfunktionen.</p></dd><dt><span class="term"><a name="gel-function-asin"></a>asin</span></dt><dd><pre
class="synopsis">asin (x)</pre><p>Alias: <code class="function">arcsin</code>
</p><p>arcsin-funktionen (invers sin).</p></dd><dt><span class="term"><a
name="gel-function-asinh"></a>asinh</span></dt><dd><pre class="synopsis">asinh (x)</pre><p>Alias: <code
class="function">arcsinh</code></p><p>arcsinh-funktionen (invers sinh).</p></dd><dt><span class="term"><a
name="gel-function-atan"></a>atan</span></dt><dd><pre class="synopsis">atan (x)</pre><p>Alias: <code
class="function">arctan</code></p><p>Beräknar arcustangensfunktionen (invers tangens).</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Arctangent"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-atanh"></a>atanh</span></dt><dd><pre
class="synopsis">atanh (x)</pre><p>Alias: <code class="function">arctanh</code></p><p>arctanh-funktionen
(invers tanh).</p></dd><dt><span class="term"><a name="gel-function-atan2"></a>atan2</span>
</dt><dd><pre class="synopsis">atan2 (y, x)</pre><p>Alias: <code
class="function">arctan2</code></p><p>Beräknar arctan2-funktionen. Om <strong
class="userinput"><code>x>0</code></strong> returnerar den <strong
class="userinput"><code>atan(y/x)</code></strong>. If <strong class="userinput"><code>x<0</code></strong>
returnerar den <strong class="userinput"><code>sign(y) * (pi - atan(|y/x|)</code></strong>. Då <strong
class="userinput"><code>x=0</code></strong> returnerar den <strong class="userinput"><code>sign(y) *
+ pi/2</code></strong>. <strong class="userinput"><code>atan2(0,0)</code></strong> returnerar 0
snarare än att misslyckas.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Atan2";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://mathworld.wolfram.com/InverseTangent.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cos"></a>cos</span></dt><dd><pre class="synopsis">cos (x)</pre><p>Beräknar
cosinusfunktionen.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Trigonometric_functions";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cosh"></a>cosh</span></dt><dd><pre class="synopsis">cosh (x)</pre><p>Beräknar funktionen
för hyperbolisk cosinus.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hyperbolic_func
tion" target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/HyperbolicFunctions";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cot"></a>cot</span></dt><dd><pre class="synopsis">cot
(x)</pre><p>Cotangensfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-coth"></a>coth</span></dt><dd><pre
class="synopsis">coth (x)</pre><p>Hyperboliska cotangensfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-csc">
</a>csc</span></dt><dd><pre class="synopsis">csc (x)</pre><p>Cosekantfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-csch"></a>csch</span></dt><dd><pre
class="synopsis">csch (x)</pre><p>Hyperboliska cosekantfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sec"></a>sec</span></dt><dd><pre
class="synopsis">sec (x)</pre><p>Sekantfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink"
href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sech"></a>sech</span></dt><dd><pre
class="synopsis">sech (x)</pre><p>Hyperboliska sekantfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sin"></a>sin</span></dt><dd><pre
class="synopsis">sin (x)</pre><p>Beräknar sinusfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-sinh"></a>sinh</span></dt><dd><pre
class="synopsis">s
inh (x)</pre><p>Beräknar funktionen för hyperbolisk sinus.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HyperbolicFunctions"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-tan"></a>tan</span></dt><dd><pre
class="synopsis">tan (x)</pre><p>Beräknar tangensfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Trigonometric_functions"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/DefinitionsInTrigonometry"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-tanh"></a>tanh</span></dt><dd><pre
class="synopsis">tanh (x)</pre><p>Hyperboliska tangensfunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hyperbolic_function"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://p
lanetmath.org/HyperbolicFunctions" target="_top">Planetmath</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s05.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s07.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Numeriska
funktioner </td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Talteori</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s07.html b/help/sv/html/ch11s07.html
index 5e13229..fe7b7c1 100644
--- a/help/sv/html/ch11s07.html
+++ b/help/sv/html/ch11s07.html
@@ -1,8 +1,8 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Talteori</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s06.html" title="Trigonometri"><link rel="next" href="ch11s08.html"
title="Matrismanipulation"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Talteori</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s06.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s08.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a name="genius-gel-function-list-number-theory"></a>Talteori</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>Är de reella heltalen <code class="varname">a</code> och
<code class="varname">b</code> relativt prima? Returnerar <code class="constant">true</code> eller <code
class="constant">false</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Coprime_integers"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/RelativelyPrime.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</
pre><p>Returnerar det <code class="varname">n</code>:e Bernoullitalet.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/BernoulliNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alias: <code class="function">CRT</code></p><p>Hitta det
<code class="varname">x</code> som löser systemet givet av vektorn <code class="varname">a</code> modulo
elementen i <code class="varname">m</code> med den kinesiska restsatsen.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/ChineseRemainderTheorem"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/Chines
eRemainderTheorem.html" target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Givet två faktoriseringar, ange faktoriseringen av
produkten.</p><p>Se <a class="link"
href="ch11s07.html#gel-function-Factorize">Factorize</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Konvertera en vektor av värden som indikerar potenser av b till ett tal.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Konvertera ett tal till en vektor av potenser för element i bas
<code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="synops
is">DiscreteLog (n,b,q)</pre><p>Hitta diskret logaritm av <code class="varname">n</code> bas <code
class="varname">b</code> i F<sub>q</sub>, den ändliga kroppen av ordning <code class="varname">q</code>, där
<code class="varname">q</code> är ett primtal, med Silver-Pohlig-Hellman-algoritmen.</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Discrete_logarithm"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Kontrollerar delbarhet (om <code class="varname">m</code> delar <code
class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)</p
re><p>Beräkna Eulers φ-funktion för <code class="varname">n</code>, det vill säga antalet heltal mellan 1
och <code class="varname">n</code> som är relativt prima till <code class="varname">n</code>.</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/TotientFunction.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Returnera <strong class="userinput"><code>n/d</code></strong> men endast om <code
class="varname">d</code> delar <code class="varname">n</code>. Om <code class="varname">d</code> inte delar
<code class="varname">n</code> kommer denna funktion returnera skräpvärden. Detta är mycket snabbare för
väldigt st
ora tal än operationen <strong class="userinput"><code>n/d</code></strong>, men självklart bara användbart
om du vet att divisionen är exakt.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize
(n)</pre><p>Returnera faktoriseringen av ett tal som en matris. Den första raden är primtalen i
faktoriseringen (inklusive 1) och den andra raden är exponenterna. Till exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>Factorize(11*11*13)</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Talteori</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s06.html" title="Trigonometri"><link rel="next" href="ch11s08.html"
title="Matrismanipulation"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084"
alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Talteori</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s06.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s08.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" s
tyle="clear: both"><a name="genius-gel-function-list-number-theory"></a>Talteori</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AreRelativelyPrime"></a>AreRelativelyPrime</span></dt><dd><pre
class="synopsis">AreRelativelyPrime (a,b)</pre><p>Är de reella heltalen <code class="varname">a</code> och
<code class="varname">b</code> relativt prima? Returnerar <code class="constant">true</code> eller <code
class="constant">false</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Coprime_integers"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/RelativelyPrime"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/RelativelyPrime.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-BernoulliNumber"></a>BernoulliNumber</span></dt><dd><pre class="synopsis">BernoulliNumber
(n)</
pre><p>Returnerar det <code class="varname">n</code>:e Bernoullitalet.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Bernoulli_number"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/BernoulliNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-ChineseRemainder"></a>ChineseRemainder</span></dt><dd><pre
class="synopsis">ChineseRemainder (a,m)</pre><p>Alias: <code class="function">CRT</code></p><p>Hitta det
<code class="varname">x</code> som löser systemet givet av vektorn <code class="varname">a</code> modulo
elementen i <code class="varname">m</code> med den kinesiska restsatsen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/ChineseRemainderTheorem"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/Chin
eseRemainderTheorem.html" target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-CombineFactorizations"></a>CombineFactorizations</span></dt><dd><pre
class="synopsis">CombineFactorizations (a,b)</pre><p>Givet två faktoriseringar, ange faktoriseringen av
produkten.</p><p>Se <a class="link"
href="ch11s07.html#gel-function-Factorize">Factorize</a>.</p></dd><dt><span class="term"><a
name="gel-function-ConvertFromBase"></a>ConvertFromBase</span></dt><dd><pre class="synopsis">ConvertFromBase
(v,b)</pre><p>Konvertera en vektor av värden som indikerar potenser av b till ett tal.</p></dd><dt><span
class="term"><a name="gel-function-ConvertToBase"></a>ConvertToBase</span></dt><dd><pre
class="synopsis">ConvertToBase (n,b)</pre><p>Konvertera ett tal till en vektor av potenser för element i bas
<code class="varname">b</code>.</p></dd><dt><span class="term"><a
name="gel-function-DiscreteLog"></a>DiscreteLog</span></dt><dd><pre class="syno
psis">DiscreteLog (n,b,q)</pre><p>Hitta diskret logaritm av <code class="varname">n</code> bas <code
class="varname">b</code> i F<sub>q</sub>, den ändliga kroppen av ordning <code class="varname">q</code>, där
<code class="varname">q</code> är ett primtal, med Silver-Pohlig-Hellman-algoritmen.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Discrete_logarithm"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/DiscreteLogarithm"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/DiscreteLogarithm.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Divides"></a>Divides</span></dt><dd><pre
class="synopsis">Divides (m,n)</pre><p>Kontrollerar delbarhet (om <code class="varname">m</code> delar <code
class="varname">n</code>).</p></dd><dt><span class="term"><a
name="gel-function-EulerPhi"></a>EulerPhi</span></dt><dd><pre class="synopsis">EulerPhi (n)
</pre><p>Beräkna Eulers φ-funktion för <code class="varname">n</code>, det vill säga antalet heltal mellan 1
och <code class="varname">n</code> som är relativt prima till <code class="varname">n</code>.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Euler_phi"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/EulerPhifunction"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/TotientFunction.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-ExactDivision"></a>ExactDivision</span></dt><dd><pre class="synopsis">ExactDivision
(n,d)</pre><p>Returnera <strong class="userinput"><code>n/d</code></strong> men endast om <code
class="varname">d</code> delar <code class="varname">n</code>. Om <code class="varname">d</code> inte delar
<code class="varname">n</code> kommer denna funktion returnera skräpvärden. Detta är mycket snabbare för
väldig
t stora tal än operationen <strong class="userinput"><code>n/d</code></strong>, men självklart bara
användbart om du vet att divisionen är exakt.</p></dd><dt><span class="term"><a
name="gel-function-Factorize"></a>Factorize</span></dt><dd><pre class="synopsis">Factorize
(n)</pre><p>Returnera faktoriseringen av ett tal som en matris. Den första raden är primtalen i
faktoriseringen (inklusive 1) och den andra raden är exponenterna. Till exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>Factorize(11*11*13)</code></strong>
=
[1 11 13
- 1 2 1]</pre><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Factorization";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Factors"></a>Factors</span></dt><dd><pre class="synopsis">Factors (n)</pre><p>Returnera
alla faktorer av <code class="varname">n</code> i en vektor. Detta inkluderar även alla
icke-primtalsfaktorer. Det inkluderar 1 och talet självt. Så för att till exempel skriva ut alla perfekta tal
(de som är summan av sina faktorer) upp till talet 1000 kan du göra följande (detta är förstås väldigt
ineffektivt) </p><pre class="programlisting">for n=1 to 1000 do (
+ 1 2 1]</pre><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Factorization";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Factors"></a>Factors</span></dt><dd><pre class="synopsis">Factors (n)</pre><p>Returnera
alla faktorer av <code class="varname">n</code> i en vektor. Detta inkluderar även alla
icke-primtalsfaktorer. Det inkluderar 1 och talet självt. Så för att till exempel skriva ut alla perfekta tal
(de som är summan av sina faktorer) upp till talet 1000 kan du göra följande (detta är förstås väldigt
ineffektivt) </p><pre class="programlisting">for n=1 to 1000 do (
if MatrixSum (Factors(n)) == 2*n then
print(n)
)
-</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,försök)</pre><p>Försök med Fermatfaktorisering av <code
class="varname">n</code> till <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, returnerar <code
class="varname">t</code> och <code class="varname">s</code> som en vektor om möjligt, annars <code
class="constant">null</code>. <code class="varname">försök</code> anger antalet försök innan vi ger
upp.</p><p>Detta är en rätt bra faktorisering om ditt tal är produkten av två faktorer som ligger väldigt
nära varandra.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Hitta det första
primitiva elementet i F<sub>q</sub>, den finita gruppen av ordning <code class="varname">q</code>. Givetvis
måste <code class="varname">q</code> vara ett primtal.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Hitta ett slumpmässigt primitivt element i
F<sub>q</sub>, den ändliga gruppen av ordning <code class="varname">q</code> (q måste vara ett
primtal).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Beräkna diskret logaritm av n bas <code class="varname">b</code> i F<sub>q</sub>, den
ändliga gruppen av ordning <code class="varname">q</code> (<code class="varname">q</code> ett primtal) med
faktorbas <code class="varname">S</code>. <code class="varname">S</code> ska vara en kolumn av primtal,
möjligen med en andra
kolumn förberäknad av <a class="link" href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Kör förberäkningssteget av <a class="link"
href="ch11s07.html#gel-function-IndexCalculus"><code class="function">IndexCalculus</code></a> för logaritmer
bas <code class="varname">b</code> i F<sub>q</sub>, den ändliga gruppen av ordning <code
class="varname">q</code> (<code class="varname">q</code> ett primtal) för faktorbasen <code
class="varname">S</code> (där <code class="varname">S</code> är en kolumnvektor av primtal). Logaritmerna
kommer vara förberäknade och returneras i den andra kolumnen.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven (
n)</pre><p>Testar om ett heltal är jämnt.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Testar om ett positivt heltal <code
class="varname">p</code> är en Mersenneprimtalsexponent. Det vill säga om 2<sup>p</sup>-1 är ett primtal. Det
gör detta genom att slå upp det i en tabell med kända värden, vilken är relativt kort. Se även <a
class="link" href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> och <a
class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>, <a class="ulink"
href="http://mathworld.wolfram.com/MersennePrime.html"; target="_top">Mathworld</a> eller <a class="ulink"
href="http://w
ww.mersenne.org/" target="_top">GIMPS</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Testar om ett rationellt tal <code class="varname">m</code> är lika med något heltal upphöjt
till <code class="varname">n</code>. Se även <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> och <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Testar om ett
heltal är udda.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Kontrollera om ett heltal är en perfekt potens, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</span>
</dt><dd><pre class="synopsis">IsPerfectSquare (n)</pre><p>Kontrollera om ett heltal är en perfekt kvadrat
av ett heltal. Talet måste vara ett reellt heltal. Negativa heltal kan givetvis aldrig vara perfekta
kvadrater av reella heltal.</p></dd><dt><span class="term"><a
name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre class="synopsis">IsPrime (n)</pre><p>Testar om
heltal är primtal. För tal mindre än 2.5e10 är svaret deterministiskt (om Riemann-hypotesen är sann). För
större tal beror sannolikheten för ett falskt positivt svar på <a class="link"
href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. Det vill säga sannolikheten för ett falskt positivt värde
är 1/4 upphöjt till <code class="function">IsPrimeMillerRabinReps</code>. Standardvärdet 22 ger en
sannolikhet på ungefär 5.7e-14.</p><p>Om <code class="constant">false</code> returneras kan du vara säker på
att talet är sammansatt.
Om du vill vara fullständigt säker på att du har ett primtal kan du använda <a class="link"
href="ch11s07.html#gel-function-MillerRabinTestSure"><code class="function">MillerRabinTestSure</code></a>
men det kan ta mycket längre tid.</p><p>Se <a class="ulink" href="http://planetmath.org/PrimeNumber";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre class="synopsis">IsPrimitiveMod
(g,q)</pre><p>Kontrollera om <code class="varname">g</code> är primitiv i F<sub>q</sub>, den finita gruppen
av ordning <code class="varname">q</code>, där <code class="varname">q</code> är ett primtal. Om <code
class="varname">q</code> inte är ett primtal kommer resultat vara felaktiga.</p></dd><dt><span
class="term"><a name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimit
iveModWithPrimeFactors</span></dt><dd><pre class="synopsis">IsPrimitiveModWithPrimeFactors
(g,q,f)</pre><p>Kontrollera om <code class="varname">g</code> är primitiv i F<sub>q</sub>, den finita gruppen
av ordning <code class="varname">q</code>, där <code class="varname">q</code> är ett primtal och <code
class="varname">f</code> är en vektor av primtalsfaktorer av <code class="varname">q</code>-1. Om <code
class="varname">q</code> inte är ett primtal kommer resultat vara felaktiga.</p></dd><dt><span
class="term"><a name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre
class="synopsis">IsPseudoprime (n,b)</pre><p>Om <code class="varname">n</code> är ett pseudoprimtal för basen
<code class="varname">b</code> men inte ett primtal, det vill säga om <strong class="userinput"><code>b^(n-1)
== 1 mod n</code></strong>. Detta anropar <a class="link"
href="ch11s07.html#gel-function-PseudoprimeTest"><code
class="function">PseudoprimeTest</code></a></p></dd><dt><
span class="term"><a name="gel-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Testa om <code class="varname">n</code> är ett starkt
pseudoprimtal för basen <code class="varname">b</code> men inte ett primtal.</p></dd><dt><span
class="term"><a name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi
(a,b)</pre><p>Alias: <code class="function">JacobiSymbol</code></p><p>Beräkna Jacobi-symbolen (a/b) (b måste
vara udda).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Alias: <code class="function">JacobiKroneckerSymbol</code></p><p>Beräkna Jacobi-symbolen (a/b)
med Kronecker-tillägget (a/2)=(2/a) när a är udda, eller (a/2)=0 när a är jämnt.</p></dd><dt><span
class="term"><a name="gel-function-LeastAbsoluteResidue"></a>LeastAbsoluteResidue</span></dt><dd><pre c
lass="synopsis">LeastAbsoluteResidue (a,n)</pre><p>Returnera residualen av <code class="varname">a</code>
mod <code class="varname">n</code> med det minsta absolutbeloppet (i intervallet -n/2 till
n/2).</p></dd><dt><span class="term"><a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre
class="synopsis">Legendre (a,p)</pre><p>Alias: <code class="function">LegendreSymbol</code></p><p>Beräkna
Legendre-symbolen (a/p).</p><p>Se <a class="ulink" href="http://planetmath.org/LegendreSymbol";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Testa om 2<sup>p</sup>-1 är ett Mersenne-primtal med Lucas-Lehmer-testet. Se även <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> och
<a class="link"
href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/LucasLhemer";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returnerar det <code class="varname">n</code>:e Lucas-talet.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span cl
ass="term"><a name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Returnera alla maximala potenser av primtalsfaktorer
för ett tal.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>En vektor av kända Mersenne-primtalsexponenter, det vill säga
en lista över positiva heltal <code class="varname">p</code> så att 2<sup>p</sup>-1 är ett primtal. Se även
<a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a> och <a
class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>, <a class="ulink" href="http://m
athworld.wolfram.com/MersennePrime.html" target="_top">Mathworld</a> eller <a class="ulink"
href="http://www.mersenne.org/"; target="_top">GIMPS</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre
class="synopsis">MillerRabinTest (n,reps)</pre><p>Använd Miller-Rabin-primalitetstestet på <code
class="varname">n</code>, <code class="varname">reps</code> gånger. Sannolikheten för falska positiva är
<strong class="userinput"><code>(1/4)^reps</code></strong>. Det är troligen vanligen bättre att använda <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a> eftersom det
är snabbare och bättre för mindre heltal.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> eller <a c
lass="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>Använd Miller-Rabin-primalitetstestet på <code
class="varname">n</code> med tillräckliga baser för att, givet den allmänna Riemann-hypotesen, resultatet ska
vara deterministiskt.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre class="synopsis">ModInvert (n,m
)</pre><p>Returnerar inversen av n mod m.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu
(n)</pre><p>Returnera Möbiusfunktionen µ(n) beräknad i <code class="varname">n</code>. Det vill säga,
returnerar 0 om <code class="varname">n</code> inte är en produkt av distinkta primtal och <strong
class="userinput"><code>(-1)^k</code></strong> om det är en produkt av <code class="varname">k</code>
distinkta primtal.</p><p>Se <a class="ulink" href="http://planetmath.org/MoebiusFunction";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre class="synopsis">NextPrime (n)</pr
e><p>Returnerar det minsta primtalet större än <code class="varname">n</code>. Negativer av primtal anses
vara primtal så för att få det föregående primtalet kan du använda <strong
class="userinput"><code>-NextPrime(-n)</code></strong>.</p><p>Denna funktion använder GMP:s <code
class="function">mpz_nextprime</code>, som i sin tur använder det probabilistiska Miller-Rabin-testet (Se
även <a class="link" href="ch11s07.html#gel-function-MillerRabinTest"><code
class="function">MillerRabinTest</code></a>). Sannolikheten för att få falska positiva går inte att ställa
in, men är låg nog för alla praktiska användningsområden.</p><p>Se <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synops
is">PadicValuation (n,p)</pre><p>Returnera den p-adiska beräkningen (antal efterföljande nollor i bas <code
class="varname">p</code>).</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/P-adic_order";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/PAdicValuation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre class="synopsis">PowerMod
(a,b,m)</pre><p>Beräkna <strong class="userinput"><code>a^b mod m</code></strong>. <code
class="varname">b</code>-potensen av <code class="varname">a</code> modulo <code class="varname">m</code>.
Det är inte nödvändigt att använda denna funktion eftersom den används automatiskt i moduloläge. Därför går
<strong class="userinput"><code>a^b mod m</code></strong> precis lika snabbt.</p></dd><dt><span
class="term"><a name="gel-function-Prime"></a>Prime</span></dt><dd><pre class="synopsis">Prime (n)
</pre><p>Alias: <code class="function">prime</code></p><p>Returnera det <code class="varname">n</code>:e
primtalet (upp till en gräns).</p><p>Se <a class="ulink" href="http://planetmath.org/PrimeNumber";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre class="synopsis">PrimeFactors
(n)</pre><p>Returnera alla primtalsfaktorer för ett tal som en vektor.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Prime_factor"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">PseudoprimeTest
(n,b)</pre><p>Pseudoprimtalstest, returnerar <code
class="constant">true</code> om och endast om <strong class="userinput"><code>b^(n-1) == 1 mod
n</code></strong></p><p>Se <a class="ulink" href="http://planetmath.org/Pseudoprime";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/Pseudoprime.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre class="synopsis">RemoveFactor
(n,m)</pre><p>Tar bort alla förekomster av faktorn <code class="varname">m</code> från talet <code
class="varname">n</code>. Det vill säga dividerar med den största potensen av <code class="varname">m</code>
som delar <code class="varname">n</code>.</p><p>Se <a class="ulink" href="http://planetmath.org/Divisibility";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/Factor.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a name="gel-fun
ction-SilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Hitta diskret logaritm av <code
class="varname">n</code> bas <code class="varname">b</code> i F<sub>q</sub>, den finita gruppen av ordning
<code class="varname">q</code>, där <code class="varname">q</code> är ett primtal med
Silver-Pohlig-Hellman-algoritmen, givet att <code class="varname">f</code> är faktoriseringen av <code
class="varname">q</code>-1.</p></dd><dt><span class="term"><a
name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Hitta kvadratrot av <code class="varname">n</code> mod <code class="varname">p</code> (där
<code class="varname">p</code> är ett primtal). Null returneras om inte en kvadratisk rest.</p><p>Se <a
class="ulink" href="http://planetmath.org/QuadraticResidue"; target="_top">Planetmath</a> eller <a class="uli
nk" href="http://mathworld.wolfram.com/QuadraticResidue.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Kör det starka pseudoprimtalstestet bas <code
class="varname">b</code> på <code class="varname">n</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Strong_pseudoprime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/StrongPseudoprime"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/StrongPseudoprime.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-gcd"></a>gcd</span></dt><dd><pre
class="synopsis">gcd (a,arg...)</pre><p>Alias: <code class="function">GCD</code></p><p>Största gemensamma
delare av heltal. Du kan mata in så många heltal som du vill i
argumentlistan, eller så kan du ange en vektor eller en matris av heltal. Om du anger mer än en matris av
samma storlek kommer SGD att utföras elementvis.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Greatest_common_divisor"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/GreatestCommonDivisor.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-lcm"></a>lcm</span></dt><dd><pre
class="synopsis">lcm (a,arg...)</pre><p>Alias: <code class="function">LCM</code></p><p>Minsta gemensamma
multipel av heltal. Du kan mata in så många heltal som du vill i argumentlistan, eller så kan du ange en
vektor eller en matris av heltal. Om du anger mer än en matris av samma storlek kommer MGM att utföras
elementvis.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Leas
t_common_multiple" target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LeastCommonMultiple"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/LeastCommonMultiple.html"; target="_top">Mathworld</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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+</pre></dd><dt><span class="term"><a
name="gel-function-FermatFactorization"></a>FermatFactorization</span></dt><dd><pre
class="synopsis">FermatFactorization (n,försök)</pre><p>Försök med Fermatfaktorisering av <code
class="varname">n</code> till <strong class="userinput"><code>(t-s)*(t+s)</code></strong>, returnerar <code
class="varname">t</code> och <code class="varname">s</code> som en vektor om möjligt, annars <code
class="constant">null</code>. <code class="varname">försök</code> anger antalet försök innan vi ger
upp.</p><p>Detta är en rätt bra faktorisering om ditt tal är produkten av två faktorer som ligger väldigt
nära varandra.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Fermat_factorization";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-FindPrimitiveElementMod"></a>FindPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindPrimitiveElementMod (q)</pre><p>Hitta det först
a primitiva elementet i F<sub>q</sub>, den finita gruppen av ordning <code class="varname">q</code>.
Givetvis måste <code class="varname">q</code> vara ett primtal.</p></dd><dt><span class="term"><a
name="gel-function-FindRandomPrimitiveElementMod"></a>FindRandomPrimitiveElementMod</span></dt><dd><pre
class="synopsis">FindRandomPrimitiveElementMod (q)</pre><p>Hitta ett slumpmässigt primitivt element i
F<sub>q</sub>, den ändliga gruppen av ordning <code class="varname">q</code> (q måste vara ett
primtal).</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculus"></a>IndexCalculus</span></dt><dd><pre class="synopsis">IndexCalculus
(n,b,q,S)</pre><p>Beräkna diskret logaritm av n bas <code class="varname">b</code> i F<sub>q</sub>, den
ändliga gruppen av ordning <code class="varname">q</code> (<code class="varname">q</code> ett primtal) med
faktorbas <code class="varname">S</code>. <code class="varname">S</code> ska vara en kolumn av primtal,
möjligen med en andr
a kolumn förberäknad av <a class="link" href="ch11s07.html#gel-function-IndexCalculusPrecalculation"><code
class="function">IndexCalculusPrecalculation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-IndexCalculusPrecalculation"></a>IndexCalculusPrecalculation</span></dt><dd><pre
class="synopsis">IndexCalculusPrecalculation (b,q,S)</pre><p>Kör förberäkningssteget av <a class="link"
href="ch11s07.html#gel-function-IndexCalculus"><code class="function">IndexCalculus</code></a> för logaritmer
bas <code class="varname">b</code> i F<sub>q</sub>, den ändliga gruppen av ordning <code
class="varname">q</code> (<code class="varname">q</code> ett primtal) för faktorbasen <code
class="varname">S</code> (där <code class="varname">S</code> är en kolumnvektor av primtal). Logaritmerna
kommer vara förberäknade och returneras i den andra kolumnen.</p></dd><dt><span class="term"><a
name="gel-function-IsEven"></a>IsEven</span></dt><dd><pre class="synopsis">IsEven
(n)</pre><p>Testar om ett heltal är jämnt.</p></dd><dt><span class="term"><a
name="gel-function-IsMersennePrimeExponent"></a>IsMersennePrimeExponent</span></dt><dd><pre
class="synopsis">IsMersennePrimeExponent (p)</pre><p>Testar om ett positivt heltal <code
class="varname">p</code> är en Mersenneprimtalsexponent. Det vill säga om 2<sup>p</sup>-1 är ett primtal. Det
gör detta genom att slå upp det i en tabell med kända värden, vilken är relativt kort. Se även <a
class="link" href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> och <a
class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>, <a class="ulink"
href="http://mathworld.wolfram.com/MersennePrime.html"; target="_top">Mathworld</a> eller <a class="ulink"
href="http:/
/www.mersenne.org/" target="_top">GIMPS</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsNthPower"></a>IsNthPower</span></dt><dd><pre class="synopsis">IsNthPower
(m,n)</pre><p>Testar om ett rationellt tal <code class="varname">m</code> är lika med något heltal upphöjt
till <code class="varname">n</code>. Se även <a class="link"
href="ch11s07.html#gel-function-IsPerfectPower">IsPerfectPower</a> och <a class="link"
href="ch11s07.html#gel-function-IsPerfectSquare">IsPerfectSquare</a>.</p></dd><dt><span class="term"><a
name="gel-function-IsOdd"></a>IsOdd</span></dt><dd><pre class="synopsis">IsOdd (n)</pre><p>Testar om ett
heltal är udda.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectPower"></a>IsPerfectPower</span></dt><dd><pre class="synopsis">IsPerfectPower
(n)</pre><p>Kontrollera om ett heltal är en perfekt potens, a<sup>b</sup>.</p></dd><dt><span class="term"><a
name="gel-function-IsPerfectSquare"></a>IsPerfectSquare</spa
n></dt><dd><pre class="synopsis">IsPerfectSquare (n)</pre><p>Kontrollera om ett heltal är en perfekt kvadrat
av ett heltal. Talet måste vara ett heltal. Negativa heltal kan givetvis aldrig vara perfekta kvadrater av
heltal.</p></dd><dt><span class="term"><a name="gel-function-IsPrime"></a>IsPrime</span></dt><dd><pre
class="synopsis">IsPrime (n)</pre><p>Testar om heltal är primtal. För tal mindre än 2.5e10 är svaret
deterministiskt (om Riemann-hypotesen är sann). För större tal beror sannolikheten för ett falskt positivt
svar på <a class="link" href="ch11s03.html#gel-function-IsPrimeMillerRabinReps"><code
class="function">IsPrimeMillerRabinReps</code></a>. Det vill säga sannolikheten för ett falskt positivt värde
är 1/4 upphöjt till <code class="function">IsPrimeMillerRabinReps</code>. Standardvärdet 22 ger en
sannolikhet på ungefär 5.7e-14.</p><p>Om <code class="constant">false</code> returneras kan du vara säker på
att talet är sammansatt. Om du vill v
ara fullständigt säker på att du har ett primtal kan du använda <a class="link"
href="ch11s07.html#gel-function-MillerRabinTestSure"><code class="function">MillerRabinTestSure</code></a>
men det kan ta mycket längre tid.</p><p>Se <a class="ulink" href="http://planetmath.org/PrimeNumber";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsPrimitiveMod"></a>IsPrimitiveMod</span></dt><dd><pre class="synopsis">IsPrimitiveMod
(g,q)</pre><p>Kontrollera om <code class="varname">g</code> är primitiv i F<sub>q</sub>, den finita gruppen
av ordning <code class="varname">q</code>, där <code class="varname">q</code> är ett primtal. Om <code
class="varname">q</code> inte är ett primtal kommer resultat vara felaktiga.</p></dd><dt><span
class="term"><a name="gel-function-IsPrimitiveModWithPrimeFactors"></a>IsPrimitiveModWithPr
imeFactors</span></dt><dd><pre class="synopsis">IsPrimitiveModWithPrimeFactors (g,q,f)</pre><p>Kontrollera
om <code class="varname">g</code> är primitiv i F<sub>q</sub>, den finita gruppen av ordning <code
class="varname">q</code>, där <code class="varname">q</code> är ett primtal och <code
class="varname">f</code> är en vektor av primtalsfaktorer av <code class="varname">q</code>-1. Om <code
class="varname">q</code> inte är ett primtal kommer resultat vara felaktiga.</p></dd><dt><span
class="term"><a name="gel-function-IsPseudoprime"></a>IsPseudoprime</span></dt><dd><pre
class="synopsis">IsPseudoprime (n,b)</pre><p>Om <code class="varname">n</code> är ett pseudoprimtal för basen
<code class="varname">b</code> men inte ett primtal, det vill säga om <strong class="userinput"><code>b^(n-1)
== 1 mod n</code></strong>. Detta anropar <a class="link"
href="ch11s07.html#gel-function-PseudoprimeTest"><code
class="function">PseudoprimeTest</code></a></p></dd><dt><span class="
term"><a name="gel-function-IsStrongPseudoprime"></a>IsStrongPseudoprime</span></dt><dd><pre
class="synopsis">IsStrongPseudoprime (n,b)</pre><p>Testa om <code class="varname">n</code> är ett starkt
pseudoprimtal för basen <code class="varname">b</code> men inte ett primtal.</p></dd><dt><span
class="term"><a name="gel-function-Jacobi"></a>Jacobi</span></dt><dd><pre class="synopsis">Jacobi
(a,b)</pre><p>Alias: <code class="function">JacobiSymbol</code></p><p>Beräkna Jacobi-symbolen (a/b) (b måste
vara udda).</p></dd><dt><span class="term"><a
name="gel-function-JacobiKronecker"></a>JacobiKronecker</span></dt><dd><pre class="synopsis">JacobiKronecker
(a,b)</pre><p>Alias: <code class="function">JacobiKroneckerSymbol</code></p><p>Beräkna Jacobi-symbolen (a/b)
med Kronecker-tillägget (a/2)=(2/a) när a är udda, eller (a/2)=0 när a är jämnt.</p></dd><dt><span
class="term"><a name="gel-function-LeastAbsoluteResidue"></a>LeastAbsoluteResidue</span></dt><dd><pre
class="synops
is">LeastAbsoluteResidue (a,n)</pre><p>Returnera residualen av <code class="varname">a</code> mod <code
class="varname">n</code> med det minsta absolutbeloppet (i intervallet -n/2 till n/2).</p></dd><dt><span
class="term"><a name="gel-function-Legendre"></a>Legendre</span></dt><dd><pre class="synopsis">Legendre
(a,p)</pre><p>Alias: <code class="function">LegendreSymbol</code></p><p>Beräkna Legendre-symbolen
(a/p).</p><p>Se <a class="ulink" href="http://planetmath.org/LegendreSymbol"; target="_top">Planetmath</a>
eller <a class="ulink" href="http://mathworld.wolfram.com/LegendreSymbol.html"; target="_top">Mathworld</a>
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-LucasLehmer"></a>LucasLehmer</span></dt><dd><pre class="synopsis">LucasLehmer
(p)</pre><p>Testa om 2<sup>p</sup>-1 är ett Mersenne-primtal med Lucas-Lehmer-testet. Se även <a class="link"
href="ch11s07.html#gel-function-MersennePrimeExponents">MersennePrimeExponents</a> och <a class="li
nk" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/LucasLhemer";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/Lucas-LehmerTest.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-LucasNumber"></a>LucasNumber</span></dt><dd><pre class="synopsis">LucasNumber
(n)</pre><p>Returnerar det <code class="varname">n</code>:e Lucas-talet.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Lucas_number"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LucasNumbers"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/LucasNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"
<a name="gel-function-MaximalPrimePowerFactors"></a>MaximalPrimePowerFactors</span></dt><dd><pre
class="synopsis">MaximalPrimePowerFactors (n)</pre><p>Returnera alla maximala potenser av primtalsfaktorer
för ett tal.</p></dd><dt><span class="term"><a
name="gel-function-MersennePrimeExponents"></a>MersennePrimeExponents</span></dt><dd><pre
class="synopsis">MersennePrimeExponents</pre><p>En vektor av kända Mersenne-primtalsexponenter, det vill
säga en lista över positiva heltal <code class="varname">p</code> så att 2<sup>p</sup>-1 är ett primtal. Se
även <a class="link" href="ch11s07.html#gel-function-IsMersennePrimeExponent">IsMersennePrimeExponent</a>
och <a class="link" href="ch11s07.html#gel-function-LucasLehmer">LucasLehmer</a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Mersenne_prime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MersenneNumbers"; target="_top">Planetmath</a>, <a class="ulink"
href="http://mathworld.
wolfram.com/MersennePrime.html" target="_top">Mathworld</a> eller <a class="ulink"
href="http://www.mersenne.org/"; target="_top">GIMPS</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-MillerRabinTest"></a>MillerRabinTest</span></dt><dd><pre
class="synopsis">MillerRabinTest (n,reps)</pre><p>Använd Miller-Rabin-primalitetstestet på <code
class="varname">n</code>, <code class="varname">reps</code> gånger. Sannolikheten för falska positiva är
<strong class="userinput"><code>(1/4)^reps</code></strong>. Det är troligen vanligen bättre att använda <a
class="link" href="ch11s07.html#gel-function-IsPrime"><code class="function">IsPrime</code></a> eftersom det
är snabbare och bättre för mindre heltal.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> eller <a
class="ul
ink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html"; target="_top">Mathworld</a>
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MillerRabinTestSure"></a>MillerRabinTestSure</span></dt><dd><pre
class="synopsis">MillerRabinTestSure (n)</pre><p>Använd Miller-Rabin-primalitetstestet på <code
class="varname">n</code> med tillräckliga baser för att, givet den allmänna Riemann-hypotesen, resultatet ska
vara deterministiskt.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"; target="_top">Wikipedia</a>, <a
class="ulink" href="http://planetmath.org/MillerRabinPrimeTest"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-ModInvert"></a>ModInvert</span></dt><dd><pre class="synopsis">ModInvert (n,m)</pre>
<p>Returnerar inversen av n mod m.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/ModularInverse.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMu"></a>MoebiusMu</span></dt><dd><pre class="synopsis">MoebiusMu
(n)</pre><p>Returnera Möbiusfunktionen µ(n) beräknad i <code class="varname">n</code>. Det vill säga,
returnerar 0 om <code class="varname">n</code> inte är en produkt av distinkta primtal och <strong
class="userinput"><code>(-1)^k</code></strong> om det är en produkt av <code class="varname">k</code>
distinkta primtal.</p><p>Se <a class="ulink" href="http://planetmath.org/MoebiusFunction";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/MoebiusFunction.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-NextPrime"></a>NextPrime</span></dt><dd><pre class="synopsis">NextPrime (n)</pre><p>Re
turnerar det minsta primtalet större än <code class="varname">n</code>. Negativer av primtal anses vara
primtal så för att få det föregående primtalet kan du använda <strong
class="userinput"><code>-NextPrime(-n)</code></strong>.</p><p>Denna funktion använder GMP:s <code
class="function">mpz_nextprime</code>, som i sin tur använder det probabilistiska Miller-Rabin-testet (Se
även <a class="link" href="ch11s07.html#gel-function-MillerRabinTest"><code
class="function">MillerRabinTest</code></a>). Sannolikheten för att få falska positiva går inte att ställa
in, men är låg nog för alla praktiska användningsområden.</p><p>Se <a class="ulink"
href="http://planetmath.org/PrimeNumber"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PrimeNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PadicValuation"></a>PadicValuation</span></dt><dd><pre class="synopsis">Pad
icValuation (n,p)</pre><p>Returnera den p-adiska beräkningen (antal efterföljande nollor i bas <code
class="varname">p</code>).</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/P-adic_order";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/PAdicValuation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-PowerMod"></a>PowerMod</span></dt><dd><pre class="synopsis">PowerMod
(a,b,m)</pre><p>Beräkna <strong class="userinput"><code>a^b mod m</code></strong>. <code
class="varname">b</code>-potensen av <code class="varname">a</code> modulo <code class="varname">m</code>.
Det är inte nödvändigt att använda denna funktion eftersom den används automatiskt i moduloläge. Därför går
<strong class="userinput"><code>a^b mod m</code></strong> precis lika snabbt.</p></dd><dt><span
class="term"><a name="gel-function-Prime"></a>Prime</span></dt><dd><pre class="synopsis">Prime (n)</pre><
p>Alias: <code class="function">prime</code></p><p>Returnera det <code class="varname">n</code>:e primtalet
(upp till en gräns).</p><p>Se <a class="ulink" href="http://planetmath.org/PrimeNumber";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/PrimeNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-PrimeFactors"></a>PrimeFactors</span></dt><dd><pre class="synopsis">PrimeFactors
(n)</pre><p>Returnera alla primtalsfaktorer för ett tal som en vektor.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Prime_factor"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PrimeFactor.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PseudoprimeTest"></a>PseudoprimeTest</span></dt><dd><pre class="synopsis">PseudoprimeTest
(n,b)</pre><p>Pseudoprimtalstest, returnerar <code class="
constant">true</code> om och endast om <strong class="userinput"><code>b^(n-1) == 1 mod
n</code></strong></p><p>Se <a class="ulink" href="http://planetmath.org/Pseudoprime";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/Pseudoprime.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RemoveFactor"></a>RemoveFactor</span></dt><dd><pre class="synopsis">RemoveFactor
(n,m)</pre><p>Tar bort alla förekomster av faktorn <code class="varname">m</code> från talet <code
class="varname">n</code>. Det vill säga dividerar med den största potensen av <code class="varname">m</code>
som delar <code class="varname">n</code>.</p><p>Se <a class="ulink" href="http://planetmath.org/Divisibility";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/Factor.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a name="gel-function-S
ilverPohligHellmanWithFactorization"></a>SilverPohligHellmanWithFactorization</span></dt><dd><pre
class="synopsis">SilverPohligHellmanWithFactorization (n,b,q,f)</pre><p>Hitta diskret logaritm av <code
class="varname">n</code> bas <code class="varname">b</code> i F<sub>q</sub>, den finita gruppen av ordning
<code class="varname">q</code>, där <code class="varname">q</code> är ett primtal med
Silver-Pohlig-Hellman-algoritmen, givet att <code class="varname">f</code> är faktoriseringen av <code
class="varname">q</code>-1.</p></dd><dt><span class="term"><a
name="gel-function-SqrtModPrime"></a>SqrtModPrime</span></dt><dd><pre class="synopsis">SqrtModPrime
(n,p)</pre><p>Hitta kvadratrot av <code class="varname">n</code> mod <code class="varname">p</code> (där
<code class="varname">p</code> är ett primtal). Null returneras om inte en kvadratisk rest.</p><p>Se <a
class="ulink" href="http://planetmath.org/QuadraticResidue"; target="_top">Planetmath</a> eller <a
class="ulink" hre
f="http://mathworld.wolfram.com/QuadraticResidue.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-StrongPseudoprimeTest"></a>StrongPseudoprimeTest</span></dt><dd><pre
class="synopsis">StrongPseudoprimeTest (n,b)</pre><p>Kör det starka pseudoprimtalstestet bas <code
class="varname">b</code> på <code class="varname">n</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Strong_pseudoprime"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/StrongPseudoprime"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/StrongPseudoprime.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-gcd"></a>gcd</span></dt><dd><pre
class="synopsis">gcd (a,arg...)</pre><p>Alias: <code class="function">GCD</code></p><p>Största gemensamma
delare av heltal. Du kan mata in så många heltal som du vill i argumen
tlistan, eller så kan du ange en vektor eller en matris av heltal. Om du anger mer än en matris av samma
storlek kommer SGD att utföras elementvis.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Greatest_common_divisor"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/GreatestCommonDivisor"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/GreatestCommonDivisor.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-lcm"></a>lcm</span></dt><dd><pre
class="synopsis">lcm (a,arg...)</pre><p>Alias: <code class="function">LCM</code></p><p>Minsta gemensamma
multipel av heltal. Du kan mata in så många heltal som du vill i argumentlistan, eller så kan du ange en
vektor eller en matris av heltal. Om du anger mer än en matris av samma storlek kommer MGM att utföras
elementvis.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Least_commo
n_multiple" target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/LeastCommonMultiple";
target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/LeastCommonMultiple.html"; target="_top">Mathworld</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s09.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 cl
ass="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Matrismanipulation</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,funk)</pre><p>Tillämpa en funktion över alla poster av en matris och returnera en matris av
resultaten.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,funk)</pre><p>Tillämpa en funktion över alla poster av två matriser
(eller ett värde och en matris) och returnera en matris av resultaten.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Hämtar
kolumnerna i en matris som en horisontell vektor.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>Compleme
ntSubmatrix</span></dt><dd><pre class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Ta bort kolumn(er) och
rad(er) från en matris.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Beräkna den k:e compound-matrisen av A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>Räkna antalet nollkolumner i en matris. Till exempel då du
kolumnreducerat en matris kan du använda detta för att hitta nulliteten. Se <a class="link"
href="ch11s09.html#gel-function-cref"><code class="function">cref</code></a> och <a class="link"
href="ch11s09.html#gel-function-Nullity"><code class="function">Nullity</code></a>.</p></dd><dt><span
class="term"><a name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre
class="synopsis">DeleteColumn (M,kol)</pre><p>Ta bort en kolumn
i en matris.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow (M,rad)</pre><p>Ta
bort en rad i en matris.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Hämtar diagonalposterna i en matris som en kolumnvektor.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Hämta skalärprodukten av två vektorer. Vektorerna måste vara av samma storlek. Inga konjugat
tas så detta är en bilinjär form även om vi arbetar över de komplexa talen; detta är den bilinjära
skalärprodukten, inte den seskvilinjära skalärprodukten. Se <a class="link"
href="ch11s08.html#gel-function-HermitianP
roduct">HermitianProduct</a> för den vanliga seskvilinjära inre produkten.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Dot_product"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/DotProduct"; target="_top">Planetmath</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre
class="synopsis">ExpandMatrix (M)</pre><p>Expanderar en matris precis som vi gör med ociterade matrisindata.
Det vill säga vi expanderar alla interna matriser som block. Detta är ett sätt att konstruera matriser från
mindre matriser och detta görs vanligen automatiskt vid inmatning om inte matrisen är
citerad.</p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre
class="synopsis">HermitianProduct (u,v)</pre><p>Alias: <code class="function">InnerProduct</code></p><p>Hämta
den hermiteska produkten av två vektorer. Vektor
erna måste vara av samma storlek. Detta är en seskvilinjär form som använder identitetsmatrisen.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Sesquilinear_form"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/HermitianInnerProduct.html"; target="_top">Mathworld</a> för
mer information.</p></dd><dt><span class="term"><a name="gel-function-I"></a>I</span></dt><dd><pre
class="synopsis">I (n)</pre><p>Alias: <code class="function">eye</code></p><p>Returnera identitetsmatris av
given storlek, det vill säga <code class="varname">n</code>×<code class="varname">n</code>. Om <code
class="varname">n</code> är noll returneras <code class="constant">null</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Identity_matrix"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-
function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vek,mstorl)</pre><p>Returnera indexkomplementet av en vektor med index. Allt är i basen ett. Till exempel
för vektorn <strong class="userinput"><code>[2,3]</code></strong> och storlek <strong
class="userinput"><code>5</code></strong> returnerar vi <strong
class="userinput"><code>[1,4,5]</code></strong>. Om <code class="varname">mstorl</code> är 0, returnerar vi
alltid <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Är
en matris diagonal.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Diagonal_matrix";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/DiagonalMatrix";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></dt>
<dd><pre class="synopsis">IsIdentity (x)</pre><p>Kontrollera om en matris är identitetsmatrisen. Returnerar
automatiskt <code class="constant">false</code> om matrisen inte är kvadratisk. Fungerar också på tal, i
vilket fall den är ekvivalent med <strong class="userinput"><code>x==1</code></strong>. Då <code
class="varname">x</code> är <code class="constant">null</code> (vi kan tänka oss detta som en 0×0-matris),
genereras inget fel och <code class="constant">false</code> returneras.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Är en matris nedåt triangulär. Det vill säga, är alla poster
ovanför diagonalen noll.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Kontrollera om en matris är en matris med heltal (icke-komplex).</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Kontrollera om en matris är icke-negativ, det vill säga om
varje element är icke-negativt. Förväxla inte positiva matriser med positivt semidefinita matriser.</p><p>Se
<a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Kontrollera om en matris är positiv, det vill säga om varje
element är positivt (och därmed reellt). Specifikt är inget element 0. Förväxla inte positiva matriser med
positivt definita matriser.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Positive_matrix";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a name="gel-function-IsMatr
ixRational"></a>IsMatrixRational</span></dt><dd><pre class="synopsis">IsMatrixRational
(M)</pre><p>Kontrollera om en matris är en matris med rationella (icke-komplexa) tal.</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre
class="synopsis">IsMatrixReal (M)</pre><p>Kontrollera om en matris är en matris med reella (icke-komplexa)
tal.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>Kontrollera om en matris är kvadratisk, det vill säga att dess bredd är samma som dess
höjd.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Är en matris uppåt triangulär? Det vill säga, en matris är
uppåt triangulär om alla poster nedanför diagonalen är noll.</p></dd><dt><span class="term"><a
name="gel-function-IsValu
eOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly (M)</pre><p>Kontrollera om en matris
är en matris med endast tal. Många interna funktioner utför denna kontroll. Värden kan vara godtyckliga tal,
inklusive komplexa tal.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Är
argument en horisontell eller vertikal vektor. Genius skiljer inte mellan en matris och en vektor, och en
vektor är bara en 1×<code class="varname">n</code>- eller <code
class="varname">n</code>×1-matrix.</p></dd><dt><span class="term"><a
name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero (x)</pre><p>Kontrollera om
en matris består av endast nollor. Fungerar också på tal, i vilket fall det är ekvivalent med <strong
class="userinput"><code>x==0</code></strong>. Då <code class="varname">x</code> är <code
class="constant">null</code> (vi kan tänka oss det som e
n 0×0-matris), genereras inget fel och <code class="constant">true</code> returneras eftersom villkoret är
tomt.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Returnerar en kopia av matrisen <code class="varname">M</code> där alla poster ovanför diagonalen
satts till noll.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Alias: <code class="function">diag</code></p><p>Skapa diagonalmatris från en vektor.
Alternativt kan du skicka med värdena att placera i diagonalen som argument. Därmed är <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> samma som <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a> eller <a class="ul
ink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Skapa en kolumnvektor från matris genom att lägga kolumner ovanpå varandra. Returnerar <code
class="constant">null</code> då den får <code class="constant">null</code> som indata.</p></dd><dt><span
class="term"><a name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre
class="synopsis">MatrixProduct (A)</pre><p>Beräkna produkten av alla element i en matris eller vektor. Det
vill säga vi multiplicerar alla element och returnerar ett tal som är produkten av alla
element.</p></dd><dt><span class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre
class="synopsis">MatrixSum (A)</pre><p>Beräkna summan av alla element i en matris eller vektor. Det vill säga
vi adderar alla element och returnerar et
t tal som är summan av alla element.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Beräkna summan av kvadraterna av alla element i en matris eller
vektor.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Returnerar en radvektor av indexen för nollskilda kolumner i matrisen <code
class="varname">M</code>.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Returnerar en radvektor av indexen för nollskilda element i vektorn <code
class="varname">v</code>.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct (u,v)</
pre><p>Hämta den yttre produkten av två vektorer. Det vill säga anta att <code class="varname">u</code> och
<code class="varname">v</code> är vertikala vektorer, då är den yttre produkten <strong
class="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Vänd på elementen i en vektor. Returnera <code class="constant">null</code> om <code
class="constant">null</code> ges</p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Beräkna summan
av varje rad i en matris och returnera en vertikal vektor med resultatet.</p></dd><dt><span class="term"><a
name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre class="synopsis">RowSumSquares
(m)</pre><p>Beräkna summan av kvadraterna för varje rad i en matris och returnera en vertikal vektor med
resulta
ten.</p></dd><dt><span class="term"><a name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre
class="synopsis">RowsOf (M)</pre><p>Hämtar raderna i en matris som en vertikal vektor. Varje element i
vektorn är en horisontell vektor som är motsvarande rad i <code class="varname">M</code>. Denna funktion är
användbar om du vill köra en slinga över raderna i en matris. Till exempel som i <strong
class="userinput"><code>for r in RowsOf(M) do
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class="titlepage"><div><div><h2 cl
ass="title" style="clear: both"><a
name="genius-gel-function-list-matrix"></a>Matrismanipulation</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-ApplyOverMatrix"></a>ApplyOverMatrix</span></dt><dd><pre class="synopsis">ApplyOverMatrix
(a,funk)</pre><p>Tillämpa en funktion över alla poster av en matris och returnera en matris av
resultaten.</p></dd><dt><span class="term"><a
name="gel-function-ApplyOverMatrix2"></a>ApplyOverMatrix2</span></dt><dd><pre
class="synopsis">ApplyOverMatrix2 (a,b,funk)</pre><p>Tillämpa en funktion över alla poster av två matriser
(eller ett värde och en matris) och returnera en matris av resultaten.</p></dd><dt><span class="term"><a
name="gel-function-ColumnsOf"></a>ColumnsOf</span></dt><dd><pre class="synopsis">ColumnsOf (M)</pre><p>Hämtar
kolumnerna i en matris som en horisontell vektor.</p></dd><dt><span class="term"><a
name="gel-function-ComplementSubmatrix"></a>Compleme
ntSubmatrix</span></dt><dd><pre class="synopsis">ComplementSubmatrix (m,r,c)</pre><p>Ta bort kolumn(er) och
rad(er) från en matris.</p></dd><dt><span class="term"><a
name="gel-function-CompoundMatrix"></a>CompoundMatrix</span></dt><dd><pre class="synopsis">CompoundMatrix
(k,A)</pre><p>Beräkna den k:e compound-matrisen av A.</p></dd><dt><span class="term"><a
name="gel-function-CountZeroColumns"></a>CountZeroColumns</span></dt><dd><pre
class="synopsis">CountZeroColumns (M)</pre><p>Räkna antalet nollkolumner i en matris. Till exempel då du
kolumnreducerat en matris kan du använda detta för att hitta nulliteten. Se <a class="link"
href="ch11s09.html#gel-function-cref"><code class="function">cref</code></a> och <a class="link"
href="ch11s09.html#gel-function-Nullity"><code class="function">Nullity</code></a>.</p></dd><dt><span
class="term"><a name="gel-function-DeleteColumn"></a>DeleteColumn</span></dt><dd><pre
class="synopsis">DeleteColumn (M,kol)</pre><p>Ta bort en kolumn
i en matris.</p></dd><dt><span class="term"><a
name="gel-function-DeleteRow"></a>DeleteRow</span></dt><dd><pre class="synopsis">DeleteRow (M,rad)</pre><p>Ta
bort en rad i en matris.</p></dd><dt><span class="term"><a
name="gel-function-DiagonalOf"></a>DiagonalOf</span></dt><dd><pre class="synopsis">DiagonalOf
(M)</pre><p>Hämtar diagonalposterna i en matris som en kolumnvektor.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-DotProduct"></a>DotProduct</span></dt><dd><pre class="synopsis">DotProduct
(u,v)</pre><p>Hämta skalärprodukten av två vektorer. Vektorerna måste vara av samma storlek. Inga konjugat
tas så detta är en bilinjär form även om vi arbetar över de komplexa talen; detta är den bilinjära
skalärprodukten, inte den seskvilinjära skalärprodukten. Se <a class="link"
href="ch11s08.html#gel-function-Hermitian
Product">HermitianProduct</a> för den vanliga seskvilinjära inre produkten.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Dot_product"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/DotProduct"; target="_top">Planetmath</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-ExpandMatrix"></a>ExpandMatrix</span></dt><dd><pre
class="synopsis">ExpandMatrix (M)</pre><p>Expanderar en matris precis som vi gör med ociterade matrisindata.
Det vill säga vi expanderar alla interna matriser som block. Detta är ett sätt att konstruera matriser från
mindre matriser och detta görs vanligen automatiskt vid inmatning om inte matrisen är
citerad.</p></dd><dt><span class="term"><a
name="gel-function-HermitianProduct"></a>HermitianProduct</span></dt><dd><pre
class="synopsis">HermitianProduct (u,v)</pre><p>Alias: <code class="function">InnerProduct</code></p><p>Hämta
den hermiteska produkten av två vektorer. Vekto
rerna måste vara av samma storlek. Detta är en seskvilinjär form som använder identitetsmatrisen.</p><p>Se
<a class="ulink" href="https://en.wikipedia.org/wiki/Sesquilinear_form"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/HermitianInnerProduct.html"; target="_top">Mathworld</a> för
mer information.</p></dd><dt><span class="term"><a name="gel-function-I"></a>I</span></dt><dd><pre
class="synopsis">I (n)</pre><p>Alias: <code class="function">eye</code></p><p>Returnera identitetsmatris av
given storlek, det vill säga <code class="varname">n</code>×<code class="varname">n</code>. Om <code
class="varname">n</code> är noll returneras <code class="constant">null</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Identity_matrix"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/IdentityMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel
-function-IndexComplement"></a>IndexComplement</span></dt><dd><pre class="synopsis">IndexComplement
(vek,mstorl)</pre><p>Returnera indexkomplementet av en vektor med index. Allt är i basen ett. Till exempel
för vektorn <strong class="userinput"><code>[2,3]</code></strong> och storlek <strong
class="userinput"><code>5</code></strong> returnerar vi <strong
class="userinput"><code>[1,4,5]</code></strong>. Om <code class="varname">mstorl</code> är 0, returnerar vi
alltid <code class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-IsDiagonal"></a>IsDiagonal</span></dt><dd><pre class="synopsis">IsDiagonal (M)</pre><p>Är
en matris diagonal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Diagonal_matrix";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/DiagonalMatrix";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsIdentity"></a>IsIdentity</span></d
t><dd><pre class="synopsis">IsIdentity (x)</pre><p>Kontrollera om en matris är identitetsmatrisen.
Returnerar automatiskt <code class="constant">false</code> om matrisen inte är kvadratisk. Fungerar också på
tal, i vilket fall den är ekvivalent med <strong class="userinput"><code>x==1</code></strong>. Då <code
class="varname">x</code> är <code class="constant">null</code> (vi kan tänka oss detta som en 0×0-matris),
genereras inget fel och <code class="constant">false</code> returneras.</p></dd><dt><span class="term"><a
name="gel-function-IsLowerTriangular"></a>IsLowerTriangular</span></dt><dd><pre
class="synopsis">IsLowerTriangular (M)</pre><p>Är en matris nedåt triangulär. Det vill säga, är alla poster
ovanför diagonalen noll.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixInteger"></a>IsMatrixInteger</span></dt><dd><pre class="synopsis">IsMatrixInteger
(M)</pre><p>Kontrollera om en matris är en matris med heltal (icke-komplex).</p></dd><dt><sp
an class="term"><a name="gel-function-IsMatrixNonnegative"></a>IsMatrixNonnegative</span></dt><dd><pre
class="synopsis">IsMatrixNonnegative (M)</pre><p>Kontrollera om en matris är icke-negativ, det vill säga om
varje element är icke-negativt. Förväxla inte positiva matriser med positivt semidefinita matriser.</p><p>Se
<a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixPositive"></a>IsMatrixPositive</span></dt><dd><pre
class="synopsis">IsMatrixPositive (M)</pre><p>Kontrollera om en matris är positiv, det vill säga om varje
element är positivt (och därmed reellt). Specifikt är inget element 0. Förväxla inte positiva matriser med
positivt definita matriser.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Positive_matrix";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a name="gel-function-Is
MatrixRational"></a>IsMatrixRational</span></dt><dd><pre class="synopsis">IsMatrixRational
(M)</pre><p>Kontrollera om en matris är en matris med rationella (icke-komplexa) tal.</p></dd><dt><span
class="term"><a name="gel-function-IsMatrixReal"></a>IsMatrixReal</span></dt><dd><pre
class="synopsis">IsMatrixReal (M)</pre><p>Kontrollera om en matris är en matris med reella (icke-komplexa)
tal.</p></dd><dt><span class="term"><a
name="gel-function-IsMatrixSquare"></a>IsMatrixSquare</span></dt><dd><pre class="synopsis">IsMatrixSquare
(M)</pre><p>Kontrollera om en matris är kvadratisk, det vill säga att dess bredd är samma som dess
höjd.</p></dd><dt><span class="term"><a
name="gel-function-IsUpperTriangular"></a>IsUpperTriangular</span></dt><dd><pre
class="synopsis">IsUpperTriangular (M)</pre><p>Är en matris uppåt triangulär? Det vill säga, en matris är
uppåt triangulär om alla poster nedanför diagonalen är noll.</p></dd><dt><span class="term"><a
name="gel-function-Is
ValueOnly"></a>IsValueOnly</span></dt><dd><pre class="synopsis">IsValueOnly (M)</pre><p>Kontrollera om en
matris är en matris med endast tal. Många interna funktioner utför denna kontroll. Värden kan vara
godtyckliga tal, inklusive komplexa tal.</p></dd><dt><span class="term"><a
name="gel-function-IsVector"></a>IsVector</span></dt><dd><pre class="synopsis">IsVector (v)</pre><p>Är
argument en horisontell eller vertikal vektor. Genius skiljer inte mellan en matris och en vektor, och en
vektor är bara en 1×<code class="varname">n</code>- eller <code
class="varname">n</code>×1-matrix.</p></dd><dt><span class="term"><a
name="gel-function-IsZero"></a>IsZero</span></dt><dd><pre class="synopsis">IsZero (x)</pre><p>Kontrollera om
en matris består av endast nollor. Fungerar också på tal, i vilket fall det är ekvivalent med <strong
class="userinput"><code>x==0</code></strong>. Då <code class="varname">x</code> är <code
class="constant">null</code> (vi kan tänka oss det s
om en 0×0-matris), genereras inget fel och <code class="constant">true</code> returneras eftersom villkoret
är tomt.</p></dd><dt><span class="term"><a
name="gel-function-LowerTriangular"></a>LowerTriangular</span></dt><dd><pre class="synopsis">LowerTriangular
(M)</pre><p>Returnerar en kopia av matrisen <code class="varname">M</code> där alla poster ovanför diagonalen
satts till noll.</p></dd><dt><span class="term"><a
name="gel-function-MakeDiagonal"></a>MakeDiagonal</span></dt><dd><pre class="synopsis">MakeDiagonal
(v,arg...)</pre><p>Alias: <code class="function">diag</code></p><p>Skapa diagonalmatris från en vektor.
Alternativt kan du skicka med värdena att placera i diagonalen som argument. Därmed är <strong
class="userinput"><code>MakeDiagonal([1,2,3])</code></strong> samma som <strong
class="userinput"><code>MakeDiagonal(1,2,3)</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Diagonal_matrix"; target="_top">Wikipedia</a> eller <a clas
s="ulink" href="http://planetmath.org/DiagonalMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-MakeVector"></a>MakeVector</span></dt><dd><pre class="synopsis">MakeVector
(A)</pre><p>Skapa en kolumnvektor från matris genom att lägga kolumner ovanpå varandra. Returnerar <code
class="constant">null</code> då den får <code class="constant">null</code> som indata.</p></dd><dt><span
class="term"><a name="gel-function-MatrixProduct"></a>MatrixProduct</span></dt><dd><pre
class="synopsis">MatrixProduct (A)</pre><p>Beräkna produkten av alla element i en matris eller vektor. Det
vill säga vi multiplicerar alla element och returnerar ett tal som är produkten av alla
element.</p></dd><dt><span class="term"><a name="gel-function-MatrixSum"></a>MatrixSum</span></dt><dd><pre
class="synopsis">MatrixSum (A)</pre><p>Beräkna summan av alla element i en matris eller vektor. Det vill säga
vi adderar alla element och returner
ar ett tal som är summan av alla element.</p></dd><dt><span class="term"><a
name="gel-function-MatrixSumSquares"></a>MatrixSumSquares</span></dt><dd><pre
class="synopsis">MatrixSumSquares (A)</pre><p>Beräkna summan av kvadraterna av alla element i en matris eller
vektor.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroColumns"></a>NonzeroColumns</span></dt><dd><pre class="synopsis">NonzeroColumns
(M)</pre><p>Returnerar en radvektor av indexen för nollskilda kolumner i matrisen <code
class="varname">M</code>.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NonzeroElements"></a>NonzeroElements</span></dt><dd><pre class="synopsis">NonzeroElements
(v)</pre><p>Returnerar en radvektor av indexen för nollskilda element i vektorn <code
class="varname">v</code>.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-OuterProduct"></a>OuterProduct</span></dt><dd><pre class="synopsis">OuterProduct (u
,v)</pre><p>Hämta den yttre produkten av två vektorer. Det vill säga anta att <code class="varname">u</code>
och <code class="varname">v</code> är vertikala vektorer, då är den yttre produkten <strong
class="userinput"><code>v * u.'</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-ReverseVector"></a>ReverseVector</span></dt><dd><pre class="synopsis">ReverseVector
(v)</pre><p>Vänd på elementen i en vektor. Returnera <code class="constant">null</code> om <code
class="constant">null</code> ges</p></dd><dt><span class="term"><a
name="gel-function-RowSum"></a>RowSum</span></dt><dd><pre class="synopsis">RowSum (m)</pre><p>Beräkna summan
av varje rad i en matris och returnera en vertikal vektor med resultatet.</p></dd><dt><span class="term"><a
name="gel-function-RowSumSquares"></a>RowSumSquares</span></dt><dd><pre class="synopsis">RowSumSquares
(m)</pre><p>Beräkna summan av kvadraterna för varje rad i en matris och returnera en vertikal vektor med re
sultaten.</p></dd><dt><span class="term"><a name="gel-function-RowsOf"></a>RowsOf</span></dt><dd><pre
class="synopsis">RowsOf (M)</pre><p>Hämtar raderna i en matris som en vertikal vektor. Varje element i
vektorn är en horisontell vektor som är motsvarande rad i <code class="varname">M</code>. Denna funktion är
användbar om du vill köra en slinga över raderna i en matris. Till exempel som i <strong
class="userinput"><code>for r in RowsOf(M) do
radfunktion(r)</code></strong>.</p></dd><dt><span class="term"><a
name="gel-function-SetMatrixSize"></a>SetMatrixSize</span></dt><dd><pre class="synopsis">SetMatrixSize
(M,rader,kolumner)</pre><p>Skapa ny matris av given storlek från en gammal. Det vill säga en ny matris kommer
returneras till vilken den gamla kopieras. Poster som inte ryms tas bort och extra utrymme fylls med nollor.
Om <code class="varname">rader</code> eller <code class="varname">kolumner</code> är noll returneras <code
class="constant">null</code>.</p></dd><dt><span class="term"><a
name="gel-function-ShuffleVector"></a>ShuffleVector</span></dt><dd><pre class="synopsis">ShuffleVector
(v)</pre><p>Flytta runt element i en vektor. Returnera <code class="constant">null</code> om <code
class="constant">null</code> ges.</p><p>Version 1.0.13 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-SortVector"></a>SortVector</span></dt><dd><pre class="synopsis">SortVector
(v)</pre><p>Sortera vektorele
ment i stigande ordning.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroColumns"></a>StripZeroColumns</span></dt><dd><pre
class="synopsis">StripZeroColumns (M)</pre><p>Ta bort alla kolumner med endast nollor i <code
class="varname">M</code>.</p></dd><dt><span class="term"><a
name="gel-function-StripZeroRows"></a>StripZeroRows</span></dt><dd><pre class="synopsis">StripZeroRows
(M)</pre><p>Ta bort alla rader med endast nollor i <code class="varname">M</code>.</p></dd><dt><span
class="term"><a name="gel-function-Submatrix"></a>Submatrix</span></dt><dd><pre class="synopsis">Submatrix
(m,r,c)</pre><p>Returnera kolumn(er) och rad(er) från en matris. Detta är ekvivalent med <strong
class="userinput"><code>m@(r,c)</code></strong>. <code class="varname">r</code> och <code
class="varname">c</code> ska vara vektorer av rader och kolumner (eller enskilda tal om endast en rad eller
kolumn behövs).</p></dd><dt><span class="term"><a name="gel-function-SwapRows"></a>SwapRo
ws</span></dt><dd><pre class="synopsis">SwapRows (m,rad1,rad2)</pre><p>Byt plats på två rader i en
matris.</p></dd><dt><span class="term"><a
name="gel-function-UpperTriangular"></a>UpperTriangular</span></dt><dd><pre class="synopsis">UpperTriangular
(M)</pre><p>Returnerar en kopia av matrisen <code class="varname">M</code> där alla poster under diagonalen
satts till noll.</p></dd><dt><span class="term"><a
name="gel-function-columns"></a>columns</span></dt><dd><pre class="synopsis">columns (M)</pre><p>Hämta
antalet kolumner i en matris.</p></dd><dt><span class="term"><a
name="gel-function-elements"></a>elements</span></dt><dd><pre class="synopsis">elements (M)</pre><p>Hämta det
totala antalet element i en matris. Detta är antalet kolumner gånger antalet rader.</p></dd><dt><span
class="term"><a name="gel-function-ones"></a>ones</span></dt><dd><pre class="synopsis">ones
(rader,kolumner...)</pre><p>Skapa en matris med ettor överallt (eller en radvektor om endast ett argu
ment ges). Returnerar <code class="constant">null</code> om antingen rader eller kolumner är
noll.</p></dd><dt><span class="term"><a name="gel-function-rows"></a>rows</span></dt><dd><pre
class="synopsis">rows (M)</pre><p>Hämta antalet rader i en matris.</p></dd><dt><span class="term"><a
name="gel-function-zeros"></a>zeros</span></dt><dd><pre class="synopsis">zeros
(rader,kolumner...)</pre><p>Skapa en matris med nollor överallt (eller en radvektor om endast ett argument
ges). Returnerar <code class="constant">null</code> om antingen rader eller kolumner är
noll.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2 c
lass="title" style="clear: both"><a name="genius-gel-function-list-linear-algebra"></a>Linjär
algebra</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Hämta hjälpenhetsmatrisen av storlek <code
class="varname">n</code>. Detta är en kvadratisk matris med bara nollor, förutom element i överdiagonalen
(i,i+1) som har värdet 1. Det är Jordanblockmatrisen med ett egenvärde som är noll.</p><p>Se <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> för mer information om
Jordans normalform.</p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm
(v,A,w)</pre><p>Ber
äkna (v,w) med avseende på den bilinjära formen given av matrisen A.</p></dd><dt><span class="term"><a
name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Returnera en funktion som beräknar två vektorer med
avseende på den bilinjära formen given av A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Alias: <code
class="function">CharPoly</code></p><p>Hämta det karakteristiska polynomet som en vektor. Det vill säga
returnera koefficienterna för polynomet med den konstanta termen först. Detta är polynomet som definieras av
<strong class="userinput"><code>det(M-xI)</code></strong>. Rötterna för detta polynom är egenvärdena för
<code class="varname">M</code>. Se även <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">
CharacteristicPolynomialFunction</a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Hämta det karakteristiska polynomet som en
funktion. Detta är polynomet som definieras av <strong class="userinput"><code>det(M-xI)</code></strong>.
Rötterna för detta polynom är egenvärdena för <code class="varname">M</code>. Se även <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a>
eller <a class="ulink" href="http://pl
anetmath.org/CharacteristicEquation" target="_top">Planetmath</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre
class="synopsis">ColumnSpace (M)</pre><p>Hämta en basmatris för kolumnrummet för en matris. Det vill säga
returnera en matris vars kolumner är basen för kolumnrummet av <code class="varname">M</code>. Det vill säga
rummet som spänns upp av kolumnerna i <code class="varname">M</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre
class="synopsis">CommutationMatrix (m, n)</pre><p>Returnera kommutationsmatrisen <strong
class="userinput"><code>K(m,n)</code></strong> som är den unika <strong
class="userinput"><code>m*n</code></strong>×<strong class="userinput"><code>m*n</code></strong
-matrisen så att <strong class="userinput"><code>K(m,n) * MakeVector(A) = MakeVector(A.')</code></strong>
för alla <code class="varname">m</code>×<code class="varname">n</code>-matriser <code
class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre
class="synopsis">CompanionMatrix (p)</pre><p>Följeslagarmatris av ett polynom (som en
vektor).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Konjugattransponatet av en matris (adjungerad matris).
Detta är det samma som <strong class="userinput"><code>.'</code></strong>-operatorn.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Conjugate_transpose"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/ConjugateTranspose"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class
="term"><a name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Alias: <code class="function">convol</code></p><p>Beräkna faltningen av två horisontella
vektorer.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Beräkna faltning av två horisontella vektorer. Returnera
resultatet som en vektor och inte adderade.</p></dd><dt><span class="term"><a
name="gel-function-CrossProduct"></a>CrossProduct</span></dt><dd><pre class="synopsis">CrossProduct
(v,w)</pre><p>CrossProduct (kryssprodukt) av två vektorer i R<sup>3</sup> som en kolumnvektor.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Cross_product"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre class="
synopsis">DeterminantalDivisorsInteger (M)</pre><p>Hämta determinantdelarna av en
heltalsmatris.</p></dd><dt><span class="term"><a
name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre class="synopsis">DirectSum
(M,N...)</pre><p>Direkt summa av matriser.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-DirectSumMatrixVector"></a>DirectSumMatrixVector</span></dt><dd><pre
class="synopsis">DirectSumMatrixVector (v)</pre><p>Direkt summa av en vektor av matriser.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a> för
mer information.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Alias: <code class="function">eig</code></p><p>Hämta egenvärdena för en kvadratisk matr
is. Fungerar för närvarande endast för upp till matriser av storlek upp till 4×4-matriser eller triangulära
matriser (för vilka egenvärdena är på diagonalen).</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multipliciteter)</pre><p>Hämta egenvektorerna för en kvadratisk matris. Hämta valfritt
även egenvärdena och deras algebraiska multipliciteter. Fungerar för närvarande endast för matriser med
stolek upp till 2×2.</p><p>Se <a class="u
link" href="http://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Tillämpa Gram-Schmidt-processen (till kolumnerna) med avseende på inre produkten given av
<code class="varname">B</code>. Om <code class="varname">B</code> inte angiven används den hermiteska
produkten. <code class="varname">B</code> kan antingen vara en seskvilinjär funktion av två argument eller så
kan det vara en som ger en seskvilinjär form. Vektorerna kommer att göras ortonormala med avseende på <code
class="varname">B</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process"; target="
_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/GramSchmidtOrthogonalization";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span></dt><dd><pre class="synopsis">HankelMatrix
(k,r)</pre><p>Hankelmatris, en matris vars antidiagonaler är konstanta. <code class="varname">k</code> är den
första raden och <code class="varname">r</code> är den sista kolumnen. Det antas att båda argumenten är
vektorer och att det sista elementet i <code class="varname">c</code> är detsamma som det första elementet i
<code class="varname">r</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hankel_matrix";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Hilbertmatris av ordning <code class="varname">n</code>.</p><p>S
e <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Image"></a>Image</span></dt><dd><pre
class="synopsis">Image (T)</pre><p>Hämta bilden (kolumnrummet) av en linjär avbildning.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre
class="synopsis">InfNorm (v)</pre><p>Hämta supremumnormen av en vektor, även kallad maximinormen eller
oändlighetsnormen.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Hämta de invarianta faktorerna för en kvadratisk
heltalsma
tris.</p></dd><dt><span class="term"><a
name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatrix</span></dt><dd><pre
class="synopsis">InverseHilbertMatrix (n)</pre><p>Invers Hilbertmatris av ordning <code
class="varname">n</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/HilbertMatrix";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Är en matris hermitesk. Det vill säga lika med sitt konjugattransponat.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hermitian_matrix"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HermitianMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-IsInSubspace"></a>IsInSubspac
e</span></dt><dd><pre class="synopsis">IsInSubspace (v,W)</pre><p>Testa om en vektor är i ett
underrum.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Är en matris (eller tal) inverterbar (En heltalsmatris är inverterbar om och endast om den är
inverterbar över heltalen).</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Är en matris (eller ett tal) inverterbar över en
kropp.</p></dd><dt><span class="term"><a name="gel-function-IsNormal"></a>IsNormal</span></dt><dd><pre
class="synopsis">IsNormal (M)</pre><p>Är <code class="varname">M</code> en normal matris. Det vill säga är
<strong class="userinput"><code>M*M' == M'*M</code></strong>.</p><p>Se <a class="ulink"
href="http://planetmath.org/NormalMatrix"; target="_top">Planetmath</a> eller <a class="ulink" href=
"http://mathworld.wolfram.com/NormalMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Är <code class="varname">M</code> en hermitesk positivt
definit matris. Det vill säga om <strong class="userinput"><code>HermitianProduct(M*v,v)</code></strong>
alltid är strikt positiv för varje vektor <code class="varname">v</code>. <code class="varname">M</code>
måste vara kvadratisk och hermitesk för att vara positivt definit. Kontrollen som utförs är att varje
principal-undermatris har en icke-negativ determinant. (Se <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Observera att vissa författare
(till exempel Mathworld) inte kräver att <code class="varname">M</code> är hermitesk, och då är villkoret på
realdelen av den inre produkten, men vi delar inte
denna åskådning. Om du vill utföra denna kontroll, se bara på den hermiteska delen av matrisen <code
class="varname">M</code> enligt följande: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Positive-definite_matrix"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Är <code class="varname">M</code> en hermitesk positivt
semidefinit matris. Det vill säga om <strong class="userinput"><code>HermitianProduct(M*v,v)</code></strong>
alltid är icke-negativ för varje vektor <code class="varname">v</code>.
<code class="varname">M</code> måste vara kvadratisk och hermitesk för att vara positivt semidefinit.
Kontrollen som utförs är att varje principal-undermatris har en icke-negativ determinant. (Se <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Observera att vissa författare
inte kräver att <code class="varname">M</code> är hermitesk, och då är villkoret på realdelen av den inre
produkten, men vi delar inte denna åskådning. Om du vill utföra denna kontroll, se bara på den hermiteska
delen av matrisen <code class="varname">M</code> enligt följande: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Se <a class="ulink"
href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-functio
n-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian (M)</pre><p>Är
en matris skevhermitesk. Det vill säga är konjugattransponatet lika med den negativa matrisen.</p><p>Se <a
class="ulink" href="http://planetmath.org/SkewHermitianMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre class="synopsis">IsUnitary (M)</pre><p>Är en
matris unitär? Det vill säga, är <strong class="userinput"><code>M'*M</code></strong> och <strong
class="userinput"><code>M*M'</code></strong> lika med identiteten.</p><p>Se <a class="ulink"
href="http://planetmath.org/UnitaryTransformation"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/UnitaryMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-JordanBlock"></a>JordanBlock</span></dt><dd><pre class="syn
opsis">JordanBlock (n,lambda)</pre><p>Alias: <code class="function">J</code></p><p>Hämta Jordanblocket som
motsvarar egenvärdet <code class="varname">lambda</code> med multiplicitet <code
class="varname">n</code>.</p><p>Se <a class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/JordanBlock.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre class="synopsis">Kernel (T)</pre><p>Hämta kärnan
(nollrummet) av en linjär avbildning.</p><p>(Se <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Alias: <code
class="function">TensorProduct</code></p><p>Beräkna Kroneckerprodukten (tensorprodukt i standar
dbas) av två matriser.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/KroneckerProduct";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre
class="synopsis">LUDecomposition (A, L, U)</pre><p>Hämta LU-faktoriseringen av <code
class="varname">A</code>, det vill säga hitta en nedåt triangulär matris och uppåt triangulär matris vilkas
produkt är <code class="varname">A</code>. Lagra resultatet i <code class="varname">L</code> och <code
class="varname">U</code> som ska vara referenser. Det returnerar <code class="constant">true</code> om det
lyckas. Anta till exempel att A är en kvadratisk matris, då kommer du efter a
tt köra: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LUDecomposition(A,&L,&U)</code></strong>
-</pre><p> ha den nedre matrisen lagrad i en variabel som kallas <code class="varname">L</code> och den övre
matrisen i en variabel som kallas <code class="varname">U</code>.</p><p>Detta är LU-faktoriseringen av en
matris, även känd som Crout- och/eller Cholesky-faktorisering. (ISBN 0-201-11577-8 pp.99-103) Den uppåt
triangulära matrisen har värdet 1 (ett) på diagonalen. Detta är inte Doolittles metod som har ettorna
diagonalt på nedermatrisen.</p><p>Alla matriser har inte LU-faktoriseringar, till exempel har <strong
class="userinput"><code>[0,1;1,0]</code></strong> inte det och denna funktion returnerar <code
class="constant">false</code> i det fallet och ställer in <code class="varname">L</code> och <code
class="varname">U</code> till <code class="constant">null</code>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath</
a> eller <a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Minor"></a>Minor</span></dt><dd><pre class="synopsis">Minor (M,i,j)</pre><p>Hämta <code
class="varname">i</code>-<code class="varname">j</code>-underdeterminanten (minoren) av en matris.</p><p>Se
<a class="ulink" href="http://planetmath.org/Minor"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Returnera kolumnerna som inte är pivotkolumnerna av en matris.</p></dd><dt><span class="term"><a
name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm (v,p...)</pre><p>Alias: <code
class="function">norm</code></p><p>Hämta p-normen (eller 2-normen om inget p är angivet) för en
vektor.</p></dd><dt><span class="term"><a nam
e="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre class="synopsis">NullSpace (T)</pre><p>Hämta
nollrummet för en matris. Det vill säga kärnan för den linjära avbildningen som matrisen representerar. Detta
returneras som en matris vars kolumnrum är nollrummet av <code class="varname">T</code>.</p><p>Se <a
class="ulink" href="http://planetmath.org/Nullspace"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre
class="synopsis">Nullity (M)</pre><p>Alias: <code class="function">nullity</code></p><p>Hämta nulliteten av
en matris. Det vill säga returnera nollrummets dimension; dimensionen på kärnan av <code
class="varname">M</code>.</p><p>Se <a class="ulink" href="http://planetmath.org/Nullity";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre class=
"synopsis">OrthogonalComplement (M)</pre><p>Hämta det ortogonala komplementet till
kolumnrummet.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Returnera pivotkolumner för en matris, det vill säga kolumner som börjar med 1 i radreducerad
trappstegsform, returnerar också raden där de förekommer.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Projicering av vektor <code class="varname">v</code> till underrum <code
class="varname">W</code> med avseende på inre produkt given av <code class="varname">B</code>. Om <code
class="varname">B</code> ej angiven används den vanliga hermiteska produkten. <code class="varname">B</code>
kan antingen vara en seskvilinjär funktion av två argument eller så kan det vara en matris som ger en
seskvilinjär form.</p></dd><dt><span class="te
rm"><a name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre
class="synopsis">QRDecomposition (A, Q)</pre><p>Hämta QR-faktoriseringen av en kvadratisk matris <code
class="varname">A</code>, returnerar den uppåt triangulära matrisen <code class="varname">R</code> och
ställer in <code class="varname">Q</code> till den ortogonala (unitära) matrisen. <code
class="varname">Q</code> bör vara en referens eller <code class="constant">null</code> om de inte vill att
något ska returneras. Till exempel: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>R = QRDecomposition(A,&Q)</code></strong>
-</pre><p> Du kommer att ha den uppåt triangulära matrisen lagrad i en variabel kallad <code
class="varname">R</code> och den ortogonala (unitära) matrisen lagrad i <code
class="varname">Q</code>.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/QRDecomposition";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Returnera Rayleighkvoten (även kallad Rayleigh-Ritz-kvoten
eller förhållandet) av en matris och en vektor.</p><p>Se <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-RayleighQu
otientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vekref)</pre><p>Hitta egenvärdena av <code
class="varname">A</code> med Rayleighkvot-iterationsmetoden. <code class="varname">x</code> är en gissning av
en egenvektor och kan vara slumpmässig. Den ska ha nollskild imaginärdel om den ska ha någon chans att hitta
komplexa egenvärden. Koden kommer köras som mest <code class="varname">maxiter</code> iterationer och
returnera <code class="constant">null</code> om vi inte kan få ett mindre fel än <code
class="varname">epsilon</code>. <code class="varname">vekref</code> ska antingen vara <code
class="constant">null</code> eller en referens till en variabel där egenvektorn ska lagras.</p><p>Se <a
class="ulink" href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> för mer information
om Rayleighkvot.</p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span>
</dt><dd><pre class="synopsis">Rank (M)</pre><p>Alias: <code class="function">rank</code></p><p>Hämta rangen
av en matris.</p><p>Se <a class="ulink" href="http://planetmath.org/SylvestersLaw";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returnerar Rossermatrisen som är ett klassiskt testproblem för symmetriska
egenvärden.</p></dd><dt><span class="term"><a
name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(vinkel)</pre><p>Alias: <code class="function">RotationMatrix</code></p><p>Returnera matrisen som motsvarar
rotation runt origo i R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(vinkel)</pre><p>Returnera matrisen som motsvarar rotation runt origo i R<sup>3</sup> kring x-axeln.</p></
dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (vinkel)</pre><p>Returnera matrisen som motsvarar rotation runt origo i
R<sup>3</sup> kring y-axeln.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(vinkel)</pre><p>Returnera matrisen som motsvarar rotation runt origo i R<sup>3</sup> kring
z-axeln.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Hämta en basmatris för radrummet av en matris.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Beräkna (v,w) med avseende på den seskvilinjära formen
given av matrisen A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFunctio
n</span></dt><dd><pre class="synopsis">SesquilinearFormFunction (A)</pre><p>Returnera en funktion som
beräknar två vektorer med avseende på den seskvilinjära formen given av A.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returnerar Smiths normalform för en matris över kroppar
(kommer i slutet ha 1:or på diagonalen).</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Smith_normal_form"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Returnerar Smiths normalform för kvadratiska
heltalsmatriser över heltal.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="t
erm"><a name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,arg...)</pre><p>Lös det linjära systemet Mx=V, returnera lösningen V
om det finns en unik lösning, returnera <code class="constant">null</code> annars. Två extra
referensparametrar kan valfritt användas för att få tag i de reducerade M och V.</p></dd><dt><span
class="term"><a name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre
class="synopsis">ToeplitzMatrix (k, r...)</pre><p>Returnera Toeplitzmatrisen skapad med den första kolumnen k
och (valfritt) den första raden r. Om endast kolumnen k anges så konjugeras den och den icke-konjugerade
versionen används som den första raden för att ge en hermitesk matris (givetvis om det första elementet är
reellt).</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Toeplitz_matrix";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/Toepli
tzMatrix" target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Trace"></a>Trace</span></dt><dd><pre class="synopsis">Trace (M)</pre><p>Alias: <code
class="function">trace</code></p><p>Beräkna spåret av en matris. Det vill säga summan av de diagonala
elementen.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/Trace";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre class="synopsis">Transpose
(M)</pre><p>Transponatet av en matris. Detta är det samma som <strong
class="userinput"><code>.'</code></strong>-operatorn.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> för mer in
formation.</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alias: <code class="function">vander</code></p><p>Returnera
Vandermondematrisen.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>Vinkeln av två vektorer med avseende på en inre produkt given av <code
class="varname">B</code>. Om <code class="varname">B</code> inte är angiven används den vanliga hermiteska
produkten. <code class="varname">B</code> kan antingen vara en seskvilinjär funktion av två argument eller så
kan det vara en matris som ger en seskvilinjär form.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum</span>
</dt><dd><pre class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Den direkta summan av vektorrummen M och
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Snitt av underrummen angivna av M och
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>Summan av vektorrummen M och N, det vill säga {w | w=m+n, m
i M, n i N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Alias: <code class="function">Adjugate</code></p><p>Hämta den klassiska
adjunkten (transponatet av kofaktormatrisen) av en matris.</p></dd><dt><span class="term"><a
name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Alias: <code
class="function">CREF</co
de><code class="function">ColumnReducedEchelonForm</code></p><p>Beräkna den kolumnreducerade
trappstegsformen.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Alias: <code class="function">Determinant</code></p><p>Hämta determinanten
av en matris.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Determinant";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/Determinant2";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-ref"></a>ref</span></dt><dd><pre class="synopsis">ref (M)</pre><p>Alias: <code
class="function">REF</code><code class="function">RowEchelonForm</code></p><p>Hämta trappstegsformen av en
matris. Det vill säga tillämpa gausselimination men inte bakåtaddition till <code class="varname">M</code>.
Pivotraderna divideras så att alla pivoter blir 1.</p><p>Se <a class="ulink" href="http://en.wikip
edia.org/wiki/Row_echelon_form" target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alias: <code class="function">RREF</code><code
class="function">ReducedRowEchelonForm</code></p><p>Hämta den radreducerade trappstegsformen av en matris.
Det vill säga tillämpa gausselimination tillsammans med bakåtaddition till <code
class="varname">M</code>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Reduced_row_echelon_form"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2 c
lass="title" style="clear: both"><a name="genius-gel-function-list-linear-algebra"></a>Linjär
algebra</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AuxiliaryUnitMatrix"></a>AuxiliaryUnitMatrix</span></dt><dd><pre
class="synopsis">AuxiliaryUnitMatrix (n)</pre><p>Hämta hjälpenhetsmatrisen av storlek <code
class="varname">n</code>. Detta är en kvadratisk matris med bara nollor, förutom element i överdiagonalen
(i,i+1) som har värdet 1. Det är Jordanblockmatrisen med ett egenvärde som är noll.</p><p>Se <a class="ulink"
href="http://planetmath.org/JordanCanonicalFormTheorem"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/JordanBlock.html"; target="_top">Mathworld</a> för mer information om
Jordans normalform.</p></dd><dt><span class="term"><a
name="gel-function-BilinearForm"></a>BilinearForm</span></dt><dd><pre class="synopsis">BilinearForm
(v,A,w)</pre><p>Ber
äkna (v,w) med avseende på den bilinjära formen given av matrisen A.</p></dd><dt><span class="term"><a
name="gel-function-BilinearFormFunction"></a>BilinearFormFunction</span></dt><dd><pre
class="synopsis">BilinearFormFunction (A)</pre><p>Returnera en funktion som beräknar två vektorer med
avseende på den bilinjära formen given av A.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomial"></a>CharacteristicPolynomial</span></dt><dd><pre
class="synopsis">CharacteristicPolynomial (M)</pre><p>Alias: <code
class="function">CharPoly</code></p><p>Hämta det karakteristiska polynomet som en vektor. Det vill säga
returnera koefficienterna för polynomet med den konstanta termen först. Detta är polynomet som definieras av
<strong class="userinput"><code>det(M-xI)</code></strong>. Rötterna för detta polynom är egenvärdena för
<code class="varname">M</code>. Se även <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomialFunction">
CharacteristicPolynomialFunction</a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/CharacteristicEquation"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-CharacteristicPolynomialFunction"></a>CharacteristicPolynomialFunction</span></dt><dd><pre
class="synopsis">CharacteristicPolynomialFunction (M)</pre><p>Hämta det karakteristiska polynomet som en
funktion. Detta är polynomet som definieras av <strong class="userinput"><code>det(M-xI)</code></strong>.
Rötterna för detta polynom är egenvärdena för <code class="varname">M</code>. Se även <a class="link"
href="ch11s09.html#gel-function-CharacteristicPolynomial">CharacteristicPolynomial</a>.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Characteristic_polynomial"; target="_top">Wikipedia</a>
eller <a class="ulink" href="http://pl
anetmath.org/CharacteristicEquation" target="_top">Planetmath</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-ColumnSpace"></a>ColumnSpace</span></dt><dd><pre
class="synopsis">ColumnSpace (M)</pre><p>Hämta en basmatris för kolumnrummet för en matris. Det vill säga
returnera en matris vars kolumner är basen för kolumnrummet av <code class="varname">M</code>. Det vill säga
rummet som spänns upp av kolumnerna i <code class="varname">M</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-CommutationMatrix"></a>CommutationMatrix</span></dt><dd><pre
class="synopsis">CommutationMatrix (m, n)</pre><p>Returnera kommutationsmatrisen <strong
class="userinput"><code>K(m,n)</code></strong> som är den unika <strong
class="userinput"><code>m*n</code></strong>×<strong class="userinput"><code>m*n</code></strong
-matrisen så att <strong class="userinput"><code>K(m,n) * MakeVector(A) = MakeVector(A.')</code></strong>
för alla <code class="varname">m</code>×<code class="varname">n</code>-matriser <code
class="varname">A</code>.</p></dd><dt><span class="term"><a
name="gel-function-CompanionMatrix"></a>CompanionMatrix</span></dt><dd><pre
class="synopsis">CompanionMatrix (p)</pre><p>Följeslagarmatris av ett polynom (som en
vektor).</p></dd><dt><span class="term"><a
name="gel-function-ConjugateTranspose"></a>ConjugateTranspose</span></dt><dd><pre
class="synopsis">ConjugateTranspose (M)</pre><p>Konjugattransponatet av en matris (adjungerad matris).
Detta är det samma som <strong class="userinput"><code>.'</code></strong>-operatorn.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Conjugate_transpose"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/ConjugateTranspose"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class
="term"><a name="gel-function-Convolution"></a>Convolution</span></dt><dd><pre class="synopsis">Convolution
(a,b)</pre><p>Alias: <code class="function">convol</code></p><p>Beräkna faltningen av två horisontella
vektorer.</p></dd><dt><span class="term"><a
name="gel-function-ConvolutionVector"></a>ConvolutionVector</span></dt><dd><pre
class="synopsis">ConvolutionVector (a,b)</pre><p>Beräkna faltning av två horisontella vektorer. Returnera
resultatet som en vektor och inte adderade.</p></dd><dt><span class="term"><a
name="gel-function-CrossProduct"></a>CrossProduct</span></dt><dd><pre class="synopsis">CrossProduct
(v,w)</pre><p>CrossProduct (kryssprodukt) av två vektorer i R<sup>3</sup> som en kolumnvektor.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Cross_product"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-DeterminantalDivisorsInteger"></a>DeterminantalDivisorsInteger</span></dt><dd><pre class="
synopsis">DeterminantalDivisorsInteger (M)</pre><p>Hämta determinantdelarna av en
heltalsmatris.</p></dd><dt><span class="term"><a
name="gel-function-DirectSum"></a>DirectSum</span></dt><dd><pre class="synopsis">DirectSum
(M,N...)</pre><p>Direkt summa av matriser.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-DirectSumMatrixVector"></a>DirectSumMatrixVector</span></dt><dd><pre
class="synopsis">DirectSumMatrixVector (v)</pre><p>Direkt summa av en vektor av matriser.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Matrix_addition#directsum"; target="_top">Wikipedia</a> för
mer information.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvalues"></a>Eigenvalues</span></dt><dd><pre class="synopsis">Eigenvalues
(M)</pre><p>Alias: <code class="function">eig</code></p><p>Hämta egenvärdena för en kvadratisk matr
is. Fungerar för närvarande endast för upp till matriser av storlek upp till 4×4-matriser eller triangulära
matriser (för vilka egenvärdena är på diagonalen).</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Eigenvalue"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/Eigenvalue"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/Eigenvalue.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Eigenvectors"></a>Eigenvectors</span></dt><dd><pre class="synopsis">Eigenvectors
(M)</pre><pre class="synopsis">Eigenvectors (M, &eigenvalues)</pre><pre class="synopsis">Eigenvectors (M,
&eigenvalues, &multipliciteter)</pre><p>Hämta egenvektorerna för en kvadratisk matris. Hämta valfritt
även egenvärdena och deras algebraiska multipliciteter. Fungerar för närvarande endast för matriser med
storlek upp till 2×2.</p><p>Se <a class=
"ulink" href="https://en.wikipedia.org/wiki/Eigenvector"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/Eigenvector"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/Eigenvector.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-GramSchmidt"></a>GramSchmidt</span></dt><dd><pre class="synopsis">GramSchmidt
(v,B...)</pre><p>Tillämpa Gram-Schmidt-processen (till kolumnerna) med avseende på inre produkten given av
<code class="varname">B</code>. Om <code class="varname">B</code> inte angiven används den hermiteska
produkten. <code class="varname">B</code> kan antingen vara en seskvilinjär funktion av två argument eller så
kan det vara en som ger en seskvilinjär form. Vektorerna kommer att göras ortonormala med avseende på <code
class="varname">B</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process"; targe
t="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/GramSchmidtOrthogonalization";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-HankelMatrix"></a>HankelMatrix</span></dt><dd><pre class="synopsis">HankelMatrix
(k,r)</pre><p>Hankelmatris, en matris vars antidiagonaler är konstanta. <code class="varname">k</code> är den
första raden och <code class="varname">r</code> är den sista kolumnen. Det antas att båda argumenten är
vektorer och att det sista elementet i <code class="varname">c</code> är detsamma som det första elementet i
<code class="varname">r</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hankel_matrix";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-HilbertMatrix"></a>HilbertMatrix</span></dt><dd><pre class="synopsis">HilbertMatrix
(n)</pre><p>Hilbertmatris av ordning <code class="varname">n</code>.</p><
p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix"; target="_top">Wikipedia</a> eller
<a class="ulink" href="http://planetmath.org/HilbertMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Image"></a>Image</span></dt><dd><pre
class="synopsis">Image (T)</pre><p>Hämta bilden (kolumnrummet) av en linjär avbildning.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Row_and_column_spaces"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-InfNorm"></a>InfNorm</span></dt><dd><pre
class="synopsis">InfNorm (v)</pre><p>Hämta supremumnormen av en vektor, även kallad maximinormen eller
oändlighetsnormen.</p></dd><dt><span class="term"><a
name="gel-function-InvariantFactorsInteger"></a>InvariantFactorsInteger</span></dt><dd><pre
class="synopsis">InvariantFactorsInteger (M)</pre><p>Hämta de invarianta faktorerna för en kvadratisk heltal
smatris.</p></dd><dt><span class="term"><a
name="gel-function-InverseHilbertMatrix"></a>InverseHilbertMatrix</span></dt><dd><pre
class="synopsis">InverseHilbertMatrix (n)</pre><p>Invers Hilbertmatris av ordning <code
class="varname">n</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Hilbert_matrix";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/HilbertMatrix";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsHermitian"></a>IsHermitian</span></dt><dd><pre class="synopsis">IsHermitian
(M)</pre><p>Är en matris hermitesk. Det vill säga lika med sitt konjugattransponat.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Hermitian_matrix"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/HermitianMatrix"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-IsInSubspace"></a>IsInSubs
pace</span></dt><dd><pre class="synopsis">IsInSubspace (v,W)</pre><p>Testa om en vektor är i ett
underrum.</p></dd><dt><span class="term"><a
name="gel-function-IsInvertible"></a>IsInvertible</span></dt><dd><pre class="synopsis">IsInvertible
(n)</pre><p>Är en matris (eller tal) inverterbar (En heltalsmatris är inverterbar om och endast om den är
inverterbar över heltalen).</p></dd><dt><span class="term"><a
name="gel-function-IsInvertibleField"></a>IsInvertibleField</span></dt><dd><pre
class="synopsis">IsInvertibleField (n)</pre><p>Är en matris (eller ett tal) inverterbar över en
kropp.</p></dd><dt><span class="term"><a name="gel-function-IsNormal"></a>IsNormal</span></dt><dd><pre
class="synopsis">IsNormal (M)</pre><p>Är <code class="varname">M</code> en normal matris. Det vill säga är
<strong class="userinput"><code>M*M' == M'*M</code></strong>.</p><p>Se <a class="ulink"
href="http://planetmath.org/NormalMatrix"; target="_top">Planetmath</a> eller <a class="ulink" hr
ef="http://mathworld.wolfram.com/NormalMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveDefinite"></a>IsPositiveDefinite</span></dt><dd><pre
class="synopsis">IsPositiveDefinite (M)</pre><p>Är <code class="varname">M</code> en hermitesk positivt
definit matris. Det vill säga om <strong class="userinput"><code>HermitianProduct(M*v,v)</code></strong>
alltid är strikt positiv för varje vektor <code class="varname">v</code>. <code class="varname">M</code>
måste vara kvadratisk och hermitesk för att vara positivt definit. Kontrollen som utförs är att varje
principal-undermatris har en icke-negativ determinant. (Se <a class="link"
href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Observera att vissa författare
(till exempel Mathworld) inte kräver att <code class="varname">M</code> är hermitesk, och då är villkoret på
realdelen av den inre produkten, men vi delar in
te denna åskådning. Om du vill utföra denna kontroll, se bara på den hermiteska delen av matrisen <code
class="varname">M</code> enligt följande: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Positive-definite_matrix"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/PositiveDefinite"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-IsPositiveSemidefinite"></a>IsPositiveSemidefinite</span></dt><dd><pre
class="synopsis">IsPositiveSemidefinite (M)</pre><p>Är <code class="varname">M</code> en hermitesk positivt
semidefinit matris. Det vill säga om <strong class="userinput"><code>HermitianProduct(M*v,v)</code></strong>
alltid är icke-negativ för varje vektor <code class="varname">v</code
. <code class="varname">M</code> måste vara kvadratisk och hermitesk för att vara positivt semidefinit.
Kontrollen som utförs är att varje principal-undermatris har en icke-negativ determinant. (Se <a
class="link" href="ch11s08.html#gel-function-HermitianProduct">HermitianProduct</a>)</p><p>Observera att
vissa författare inte kräver att <code class="varname">M</code> är hermitesk, och då är villkoret på
realdelen av den inre produkten, men vi delar inte denna åskådning. Om du vill utföra denna kontroll, se
bara på den hermiteska delen av matrisen <code class="varname">M</code> enligt följande: <strong
class="userinput"><code>IsPositiveSemidefinite(M+M')</code></strong>.</p><p>Se <a class="ulink"
href="http://planetmath.org/PositiveSemidefinite"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-func
tion-IsSkewHermitian"></a>IsSkewHermitian</span></dt><dd><pre class="synopsis">IsSkewHermitian
(M)</pre><p>Är en matris skevhermitesk. Det vill säga är konjugattransponatet lika med den negativa
matrisen.</p><p>Se <a class="ulink" href="http://planetmath.org/SkewHermitianMatrix";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-IsUnitary"></a>IsUnitary</span></dt><dd><pre class="synopsis">IsUnitary (M)</pre><p>Är en
matris unitär? Det vill säga, är <strong class="userinput"><code>M'*M</code></strong> och <strong
class="userinput"><code>M*M'</code></strong> lika med identiteten.</p><p>Se <a class="ulink"
href="http://planetmath.org/UnitaryTransformation"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/UnitaryMatrix.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-JordanBlock"></a>JordanBlock</span></dt><dd><pre class="
synopsis">JordanBlock (n,lambda)</pre><p>Alias: <code class="function">J</code></p><p>Hämta Jordanblocket
som motsvarar egenvärdet <code class="varname">lambda</code> med multiplicitet <code
class="varname">n</code>.</p><p>Se <a class="ulink" href="http://planetmath.org/JordanCanonicalFormTheorem";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/JordanBlock.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Kernel"></a>Kernel</span></dt><dd><pre class="synopsis">Kernel (T)</pre><p>Hämta kärnan
(nollrummet) av en linjär avbildning.</p><p>(Se <a class="link"
href="ch11s09.html#gel-function-NullSpace">NullSpace</a>)</p></dd><dt><span class="term"><a
name="gel-function-KroneckerProduct"></a>KroneckerProduct</span></dt><dd><pre
class="synopsis">KroneckerProduct (M, N)</pre><p>Alias: <code
class="function">TensorProduct</code></p><p>Beräkna Kroneckerprodukten (tensorprodukt i stan
dardbas) av två matriser.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Kronecker_product";
target="_top">Wikipedia</a>, <a class="ulink" href="http://planetmath.org/KroneckerProduct";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/KroneckerProduct.html";
target="_top">Mathworld</a> för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-LUDecomposition"></a>LUDecomposition</span></dt><dd><pre
class="synopsis">LUDecomposition (A, L, U)</pre><p>Hämta LU-faktoriseringen av <code
class="varname">A</code>, det vill säga hitta en nedåt triangulär matris och uppåt triangulär matris vilkas
produkt är <code class="varname">A</code>. Lagra resultatet i <code class="varname">L</code> och <code
class="varname">U</code> som ska vara referenser. Det returnerar <code class="constant">true</code> om det
lyckas. Anta till exempel att A är en kvadratisk matris, då kommer du eft
er att köra: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LUDecomposition(A,&L,&U)</code></strong>
+</pre><p> ha den nedre matrisen lagrad i en variabel som kallas <code class="varname">L</code> och den övre
matrisen i en variabel som kallas <code class="varname">U</code>.</p><p>Detta är LU-faktoriseringen av en
matris, även känd som Crout- och/eller Cholesky-faktorisering. (ISBN 0-201-11577-8 pp.99-103) Den uppåt
triangulära matrisen har värdet 1 (ett) på diagonalen. Detta är inte Doolittles metod som har ettorna
diagonalt på nedermatrisen.</p><p>Alla matriser har inte LU-faktoriseringar, till exempel har <strong
class="userinput"><code>[0,1;1,0]</code></strong> inte det och denna funktion returnerar <code
class="constant">false</code> i det fallet och ställer in <code class="varname">L</code> och <code
class="varname">U</code> till <code class="constant">null</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/LU_decomposition"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/LUDecomposition"; target="_top">Planetmath<
/a> eller <a class="ulink" href="http://mathworld.wolfram.com/LUDecomposition.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Minor"></a>Minor</span></dt><dd><pre class="synopsis">Minor (M,i,j)</pre><p>Hämta <code
class="varname">i</code>-<code class="varname">j</code>-underdeterminanten (minoren) av en matris.</p><p>Se
<a class="ulink" href="http://planetmath.org/Minor"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-NonPivotColumns"></a>NonPivotColumns</span></dt><dd><pre class="synopsis">NonPivotColumns
(M)</pre><p>Returnera kolumnerna som inte är pivotkolumnerna av en matris.</p></dd><dt><span class="term"><a
name="gel-function-Norm"></a>Norm</span></dt><dd><pre class="synopsis">Norm (v,p...)</pre><p>Alias: <code
class="function">norm</code></p><p>Hämta p-normen (eller 2-normen om inget p är angivet) för en
vektor.</p></dd><dt><span class="term"><a na
me="gel-function-NullSpace"></a>NullSpace</span></dt><dd><pre class="synopsis">NullSpace (T)</pre><p>Hämta
nollrummet för en matris. Det vill säga kärnan för den linjära avbildningen som matrisen representerar. Detta
returneras som en matris vars kolumnrum är nollrummet av <code class="varname">T</code>.</p><p>Se <a
class="ulink" href="http://planetmath.org/Nullspace"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Nullity"></a>Nullity</span></dt><dd><pre
class="synopsis">Nullity (M)</pre><p>Alias: <code class="function">nullity</code></p><p>Hämta nulliteten av
en matris. Det vill säga returnera nollrummets dimension; dimensionen på kärnan av <code
class="varname">M</code>.</p><p>Se <a class="ulink" href="http://planetmath.org/Nullity";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-OrthogonalComplement"></a>OrthogonalComplement</span></dt><dd><pre class
="synopsis">OrthogonalComplement (M)</pre><p>Hämta det ortogonala komplementet till
kolumnrummet.</p></dd><dt><span class="term"><a
name="gel-function-PivotColumns"></a>PivotColumns</span></dt><dd><pre class="synopsis">PivotColumns
(M)</pre><p>Returnera pivotkolumner för en matris, det vill säga kolumner som börjar med 1 i radreducerad
trappstegsform, returnerar också raden där de förekommer.</p></dd><dt><span class="term"><a
name="gel-function-Projection"></a>Projection</span></dt><dd><pre class="synopsis">Projection
(v,W,B...)</pre><p>Projicering av vektor <code class="varname">v</code> till underrum <code
class="varname">W</code> med avseende på inre produkt given av <code class="varname">B</code>. Om <code
class="varname">B</code> ej angiven används den vanliga hermiteska produkten. <code class="varname">B</code>
kan antingen vara en seskvilinjär funktion av två argument eller så kan det vara en matris som ger en
seskvilinjär form.</p></dd><dt><span class="t
erm"><a name="gel-function-QRDecomposition"></a>QRDecomposition</span></dt><dd><pre
class="synopsis">QRDecomposition (A, Q)</pre><p>Hämta QR-faktoriseringen av en kvadratisk matris <code
class="varname">A</code>, returnerar den uppåt triangulära matrisen <code class="varname">R</code> och
ställer in <code class="varname">Q</code> till den ortogonala (unitära) matrisen. <code
class="varname">Q</code> bör vara en referens eller <code class="constant">null</code> om de inte vill att
något ska returneras. Till exempel: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>R = QRDecomposition(A,&Q)</code></strong>
+</pre><p> Du kommer att ha den uppåt triangulära matrisen lagrad i en variabel kallad <code
class="varname">R</code> och den ortogonala (unitära) matrisen lagrad i <code
class="varname">Q</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/QR_decomposition";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/QRDecomposition";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/QRDecomposition.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RayleighQuotient"></a>RayleighQuotient</span></dt><dd><pre
class="synopsis">RayleighQuotient (A,x)</pre><p>Returnera Rayleighkvoten (även kallad Rayleigh-Ritz-kvoten
eller förhållandet) av en matris och en vektor.</p><p>Se <a class="ulink"
href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-RayleighQ
uotientIteration"></a>RayleighQuotientIteration</span></dt><dd><pre
class="synopsis">RayleighQuotientIteration (A,x,epsilon,maxiter,vekref)</pre><p>Hitta egenvärdena av <code
class="varname">A</code> med Rayleighkvot-iterationsmetoden. <code class="varname">x</code> är en gissning av
en egenvektor och kan vara slumpmässig. Den ska ha nollskild imaginärdel om den ska ha någon chans att hitta
komplexa egenvärden. Koden kommer köras som mest <code class="varname">maxiter</code> iterationer och
returnera <code class="constant">null</code> om vi inte kan få ett mindre fel än <code
class="varname">epsilon</code>. <code class="varname">vekref</code> ska antingen vara <code
class="constant">null</code> eller en referens till en variabel där egenvektorn ska lagras.</p><p>Se <a
class="ulink" href="http://planetmath.org/RayleighQuotient"; target="_top">Planetmath</a> för mer information
om Rayleighkvot.</p></dd><dt><span class="term"><a name="gel-function-Rank"></a>Rank</span
</dt><dd><pre class="synopsis">Rank (M)</pre><p>Alias: <code class="function">rank</code></p><p>Hämta
rangen av en matris.</p><p>Se <a class="ulink" href="http://planetmath.org/SylvestersLaw";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RosserMatrix"></a>RosserMatrix</span></dt><dd><pre class="synopsis">RosserMatrix
()</pre><p>Returnerar Rossermatrisen som är ett klassiskt testproblem för symmetriska
egenvärden.</p></dd><dt><span class="term"><a
name="gel-function-Rotation2D"></a>Rotation2D</span></dt><dd><pre class="synopsis">Rotation2D
(vinkel)</pre><p>Alias: <code class="function">RotationMatrix</code></p><p>Returnera matrisen som motsvarar
rotation runt origo i R<sup>2</sup>.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DX"></a>Rotation3DX</span></dt><dd><pre class="synopsis">Rotation3DX
(vinkel)</pre><p>Returnera matrisen som motsvarar rotation runt origo i R<sup>3</sup> kring x-axeln.</p><
/dd><dt><span class="term"><a name="gel-function-Rotation3DY"></a>Rotation3DY</span></dt><dd><pre
class="synopsis">Rotation3DY (vinkel)</pre><p>Returnera matrisen som motsvarar rotation runt origo i
R<sup>3</sup> kring y-axeln.</p></dd><dt><span class="term"><a
name="gel-function-Rotation3DZ"></a>Rotation3DZ</span></dt><dd><pre class="synopsis">Rotation3DZ
(vinkel)</pre><p>Returnera matrisen som motsvarar rotation runt origo i R<sup>3</sup> kring
z-axeln.</p></dd><dt><span class="term"><a name="gel-function-RowSpace"></a>RowSpace</span></dt><dd><pre
class="synopsis">RowSpace (M)</pre><p>Hämta en basmatris för radrummet av en matris.</p></dd><dt><span
class="term"><a name="gel-function-SesquilinearForm"></a>SesquilinearForm</span></dt><dd><pre
class="synopsis">SesquilinearForm (v,A,w)</pre><p>Beräkna (v,w) med avseende på den seskvilinjära formen
given av matrisen A.</p></dd><dt><span class="term"><a
name="gel-function-SesquilinearFormFunction"></a>SesquilinearFormFuncti
on</span></dt><dd><pre class="synopsis">SesquilinearFormFunction (A)</pre><p>Returnera en funktion som
beräknar två vektorer med avseende på den seskvilinjära formen given av A.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormField"></a>SmithNormalFormField</span></dt><dd><pre
class="synopsis">SmithNormalFormField (A)</pre><p>Returnerar Smiths normalform för en matris över kroppar
(kommer i slutet ha 1:or på diagonalen).</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Smith_normal_form"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-SmithNormalFormInteger"></a>SmithNormalFormInteger</span></dt><dd><pre
class="synopsis">SmithNormalFormInteger (M)</pre><p>Returnerar Smiths normalform för kvadratiska
heltalsmatriser över heltal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Smith_normal_form";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class
="term"><a name="gel-function-SolveLinearSystem"></a>SolveLinearSystem</span></dt><dd><pre
class="synopsis">SolveLinearSystem (M,V,arg...)</pre><p>Lös det linjära systemet Mx=V, returnera lösningen V
om det finns en unik lösning, returnera <code class="constant">null</code> annars. Två extra
referensparametrar kan valfritt användas för att få tag i de reducerade M och V.</p></dd><dt><span
class="term"><a name="gel-function-ToeplitzMatrix"></a>ToeplitzMatrix</span></dt><dd><pre
class="synopsis">ToeplitzMatrix (k, r...)</pre><p>Returnera Toeplitzmatrisen skapad med den första kolumnen k
och (valfritt) den första raden r. Om endast kolumnen k anges så konjugeras den och den icke-konjugerade
versionen används som den första raden för att ge en hermitesk matris (givetvis om det första elementet är
reellt).</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Toeplitz_matrix";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/To
eplitzMatrix" target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Trace"></a>Trace</span></dt><dd><pre class="synopsis">Trace (M)</pre><p>Alias: <code
class="function">trace</code></p><p>Beräkna spåret av en matris. Det vill säga summan av de diagonala
elementen.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)"
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/Trace";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Transpose"></a>Transpose</span></dt><dd><pre class="synopsis">Transpose
(M)</pre><p>Transponatet av en matris. Detta är det samma som <strong
class="userinput"><code>.'</code></strong>-operatorn.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Transpose"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/Transpose"; target="_top">Planetmath</a> för
mer information.</p></dd><dt><span class="term"><a
name="gel-function-VandermondeMatrix"></a>VandermondeMatrix</span></dt><dd><pre
class="synopsis">VandermondeMatrix (v)</pre><p>Alias: <code class="function">vander</code></p><p>Returnera
Vandermondematrisen.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Vandermonde_matrix";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-VectorAngle"></a>VectorAngle</span></dt><dd><pre class="synopsis">VectorAngle
(v,w,B...)</pre><p>Vinkeln av två vektorer med avseende på en inre produkt given av <code
class="varname">B</code>. Om <code class="varname">B</code> inte är angiven används den vanliga hermiteska
produkten. <code class="varname">B</code> kan antingen vara en seskvilinjär funktion av två argument eller så
kan det vara en matris som ger en seskvilinjär form.</p></dd><dt><span class="term"><a
name="gel-function-VectorSpaceDirectSum"></a>VectorSpaceDirectSum
</span></dt><dd><pre class="synopsis">VectorSpaceDirectSum (M,N)</pre><p>Den direkta summan av vektorrummen
M och N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceIntersection"></a>VectorSubspaceIntersection</span></dt><dd><pre
class="synopsis">VectorSubspaceIntersection (M,N)</pre><p>Snitt av underrummen angivna av M och
N.</p></dd><dt><span class="term"><a
name="gel-function-VectorSubspaceSum"></a>VectorSubspaceSum</span></dt><dd><pre
class="synopsis">VectorSubspaceSum (M,N)</pre><p>Summan av vektorrummen M och N, det vill säga {w | w=m+n, m
i M, n i N}.</p></dd><dt><span class="term"><a name="gel-function-adj"></a>adj</span></dt><dd><pre
class="synopsis">adj (m)</pre><p>Alias: <code class="function">Adjugate</code></p><p>Hämta den klassiska
adjunkten (transponatet av kofaktormatrisen) av en matris.</p></dd><dt><span class="term"><a
name="gel-function-cref"></a>cref</span></dt><dd><pre class="synopsis">cref (M)</pre><p>Alias: <code
class="function">C
REF</code><code class="function">ColumnReducedEchelonForm</code></p><p>Beräkna den kolumnreducerade
trappstegsformen.</p></dd><dt><span class="term"><a name="gel-function-det"></a>det</span></dt><dd><pre
class="synopsis">det (M)</pre><p>Alias: <code class="function">Determinant</code></p><p>Hämta determinanten
av en matris.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Determinant";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/Determinant2";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-ref"></a>ref</span></dt><dd><pre class="synopsis">ref (M)</pre><p>Alias: <code
class="function">REF</code><code class="function">RowEchelonForm</code></p><p>Hämta trappstegsformen av en
matris. Det vill säga tillämpa gausselimination men inte bakåtaddition till <code class="varname">M</code>.
Pivotraderna divideras så att alla pivoter blir 1.</p><p>Se <a class="ulink" href="https:/
/en.wikipedia.org/wiki/Row_echelon_form" target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/RowEchelonForm"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-rref"></a>rref</span></dt><dd><pre
class="synopsis">rref (M)</pre><p>Alias: <code class="function">RREF</code><code
class="function">ReducedRowEchelonForm</code></p><p>Hämta den radreducerade trappstegsformen av en matris.
Det vill säga tillämpa gausselimination tillsammans med bakåtaddition till <code
class="varname">M</code>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Reduced_row_echelon_form"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/ReducedRowEchelonForm"; target="_top">Planetmath</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a
name="genius-gel-function-list-combinatorics"></a>Kombinatorik</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Hämta det
<code class="varname">n</code>:e Catalantalet.</p><p>Se <a class="ulink"
href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Hämta alla kombinationer av k tal från 1 till n som en vektor av vektorer. (Se även <a
class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p></dd><dt><span
class="term"><a name="gel-function-DoubleFactorial"></a>DoubleFactorial</span></dt><dd><pre
class="synopsis">DoubleFactorial (n)</pre><p>Semifakultet: <strong class="userinput"><code>n(n-2)(n
-4)...</code></strong></p><p>Se <a class="ulink" href="http://planetmath.org/DoubleFactorial";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre class="synopsis">Factorial
(n)</pre><p>Fakultet: <strong class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>Se <a
class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre
class="synopsis">FallingFactorial (n,k)</pre><p>Fallande fakultet: <strong class="userinput"><code>(n)_k =
n(n-1)...(n-(k-1))</code></strong></p><p>Se <a class="ulink" href="http://planetmath.org/FallingFactorial";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre class="synopsis">Fibonacci (x)</pre><p>Alia
s: <code class="function">fib</code></p><p>Beräkna det <code class="varname">n</code>:e Fibonaccitalet. Det
vill säga numret som definieras rekursivt av <strong class="userinput"><code>Fibonacci(n) = Fibonacci(n-1) +
Fibonacci(n-2)</code></strong> och <strong class="userinput"><code>Fibonacci(1) = Fibonacci(2) =
1</code></strong>.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Fibonacci_number";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/FibonacciSequence";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/FibonacciNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>Beräkna Frobeniustalet. Det vill säga beräkna det största tal som inte kan anges som en
icke-negativ linjär heltalskombination av en given vektor av ic
ke-negativa tal. Vektorn kan ges som separata tal eller en ensam vektor. Alla angivna tal ska ha SGD
1.</p><p>Se <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(komberingsregel)</pre><p>Galois-matris givet en linjär kombineringsregel
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Hitta vektorn <code class="varname">c</code> av icke-negativa heltal så att skalärprodukten med
<code class="varname">v</code> är lika med n. Om inte möjligt returneras <code class="constant">null</code>.
<code class="varname">v</code> bör anges sorterad i ökande ordning och bestå av icke-negativa
heltal.</p><p>Se <a class="ulink" href="http://ma
thworld.wolfram.com/GreedyAlgorithm.html" target="_top">Mathworld</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre
class="synopsis">HarmonicNumber (n,r)</pre><p>Alias: <code class="function">HarmonicH</code></p><p>Harmoniskt
tal, det <code class="varname">n</code>:e harmoniska talet av ordning <code
class="varname">r</code>.</p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadters funktion q(n) definierad av q(1)=1, q(2)=1,
q(n)=q(n-q(n-1))+q(n-q(n-2)).</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (frövärden,kombineringsregel,n)</pre><p>Beräkna linjär rekursiv
sekvens med Galois-stegning.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span
</dt><dd><pre class="synopsis">Multinomial (v,arg...)</pre><p>Beräkna multinomialkoefficienter. Tar en
vektor av <code class="varname">k</code> icke-negativa heltal och beräknar multinomialkoefficienten. Denna
motsvarar koefficienten i det homogena polynomet i <code class="varname">k</code> variabler med motsvarande
potenser.</p><p>Formeln för <strong class="userinput"><code>Multinomial(a,b,c)</code></strong> kan skrivas
som: </p><pre class="programlisting">(a+b+c)! / (a!b!c!)
-</pre><p> Med andra ord, om vi bara skulle ha två element så är <strong
class="userinput"><code>Multinomial(a,b)</code></strong> samma sak som <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> eller <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Multinomial_theorem"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/MultinomialCoefficient.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Hämta kombination som skulle komma efter v i anrop till kombinationer, första kombination
skulle vara <strong class="userinput"><code>[1:k]</code></strong>. Denna funktion är användbar om du har
många kom
binationer att gå igenom och du inte vill slösa minne med att lagra dem alla.</p><p>Till exempel med
Combinations skulle du vanligen skriva en slinga som: </p><pre class="screen"><strong
class="userinput"><code>for n in Combinations (4,6) do (
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Kombinatorik</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s09.html" title="Linjär
algebra"><link rel="next" href="ch11s11.html" title="Kalkyl"></head><body bgcolor="white" text="black"
link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation
header"><tr><th colspan="3" align="center">Kombinatorik</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s09.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s11.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" st
yle="clear: both"><a
name="genius-gel-function-list-combinatorics"></a>Kombinatorik</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Catalan"></a>Catalan</span></dt><dd><pre class="synopsis">Catalan (n)</pre><p>Hämta det
<code class="varname">n</code>:e Catalantalet.</p><p>Se <a class="ulink"
href="http://planetmath.org/CatalanNumbers"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Combinations"></a>Combinations</span></dt><dd><pre class="synopsis">Combinations
(k,n)</pre><p>Hämta alla kombinationer av k tal från 1 till n som en vektor av vektorer. (Se även <a
class="link" href="ch11s10.html#gel-function-NextCombination">NextCombination</a>)</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-DoubleFactorial"></a>DoubleFact
orial</span></dt><dd><pre class="synopsis">DoubleFactorial (n)</pre><p>Semifakultet: <strong
class="userinput"><code>n(n-2)(n-4)...</code></strong></p><p>Se <a class="ulink"
href="http://planetmath.org/DoubleFactorial"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Factorial"></a>Factorial</span></dt><dd><pre class="synopsis">Factorial
(n)</pre><p>Fakultet: <strong class="userinput"><code>n(n-1)(n-2)...</code></strong></p><p>Se <a
class="ulink" href="http://planetmath.org/Factorial"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-FallingFactorial"></a>FallingFactorial</span></dt><dd><pre
class="synopsis">FallingFactorial (n,k)</pre><p>Fallande fakultet: <strong class="userinput"><code>(n)_k =
n(n-1)...(n-(k-1))</code></strong></p><p>Se <a class="ulink" href="http://planetmath.org/FallingFactorial";
target="_top">Planetmath</a> för mer information.</p></dd><dt><spa
n class="term"><a name="gel-function-Fibonacci"></a>Fibonacci</span></dt><dd><pre class="synopsis">Fibonacci
(x)</pre><p>Alias: <code class="function">fib</code></p><p>Beräkna det <code class="varname">n</code>:e
Fibonaccitalet. Det vill säga numret som definieras rekursivt av <strong class="userinput"><code>Fibonacci(n)
= Fibonacci(n-1) + Fibonacci(n-2)</code></strong> och <strong class="userinput"><code>Fibonacci(1) =
Fibonacci(2) = 1</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fibonacci_number"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://planetmath.org/FibonacciSequence"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FibonacciNumber.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-FrobeniusNumber"></a>FrobeniusNumber</span></dt><dd><pre class="synopsis">FrobeniusNumber
(v,arg...)</pre><p>Beräkna Frobeniustalet. Det v
ill säga beräkna det största tal som inte kan anges som en icke-negativ linjär heltalskombination av en
given vektor av icke-negativa tal. Vektorn kan ges som separata tal eller en ensam vektor. Alla angivna tal
ska ha SGD 1.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Coin_problem";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://mathworld.wolfram.com/FrobeniusNumber.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-GaloisMatrix"></a>GaloisMatrix</span></dt><dd><pre class="synopsis">GaloisMatrix
(kombineringsregel)</pre><p>Galois-matris givet en linjär kombineringsregel
(a_1*x_1+...+a_n*x_n=x_(n+1)).</p></dd><dt><span class="term"><a
name="gel-function-GreedyAlgorithm"></a>GreedyAlgorithm</span></dt><dd><pre class="synopsis">GreedyAlgorithm
(n,v)</pre><p>Hitta vektorn <code class="varname">c</code> av icke-negativa heltal så att skalärprodukten med
<code class="varname">v</cod
e> är lika med n. Om inte möjligt returneras <code class="constant">null</code>. <code
class="varname">v</code> bör anges sorterad i ökande ordning och bestå av icke-negativa heltal.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Greedy_algorithm"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/GreedyAlgorithm.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-HarmonicNumber"></a>HarmonicNumber</span></dt><dd><pre class="synopsis">HarmonicNumber
(n,r)</pre><p>Alias: <code class="function">HarmonicH</code></p><p>Harmoniskt tal, det <code
class="varname">n</code>:e harmoniska talet av ordning <code class="varname">r</code>. Det vill säga summan
av <strong class="userinput"><code>1/k^r</code></strong> för <code class="varname">k</code> från 1 till n.
Ekvivalent med <strong class="userinput"><code>sum k = 1 to n do 1/k^r</code></strong>.</p><p>Se <a
class="ulink
" href="https://en.wikipedia.org/wiki/Harmonic_number"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Hofstadter"></a>Hofstadter</span></dt><dd><pre class="synopsis">Hofstadter
(n)</pre><p>Hofstadters funktion q(n) definierad av q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Hofstadter_sequence"; target="_top">Wikipedia</a> för mer
information. Sekvensen är <a class="ulink" href="https://oeis.org/A005185"; target="_top">A005185 i
OEIS</a>.</p></dd><dt><span class="term"><a
name="gel-function-LinearRecursiveSequence"></a>LinearRecursiveSequence</span></dt><dd><pre
class="synopsis">LinearRecursiveSequence (frövärden,kombineringsregel,n)</pre><p>Beräkna linjär rekursiv
sekvens med Galois-stegning.</p></dd><dt><span class="term"><a
name="gel-function-Multinomial"></a>Multinomial</span></dt><dd><pre class="synopsis">Multinomial
(v,arg...)</pre><p>Beräkna multi
nomialkoefficienter. Tar en vektor av <code class="varname">k</code> icke-negativa heltal och beräknar
multinomialkoefficienten. Denna motsvarar koefficienten i det homogena polynomet i <code
class="varname">k</code> variabler med motsvarande potenser.</p><p>Formeln för <strong
class="userinput"><code>Multinomial(a,b,c)</code></strong> kan skrivas som: </p><pre
class="programlisting">(a+b+c)! / (a!b!c!)
+</pre><p> Med andra ord, om vi bara skulle ha två element så är <strong
class="userinput"><code>Multinomial(a,b)</code></strong> samma sak som <strong
class="userinput"><code>Binomial(a+b,a)</code></strong> eller <strong
class="userinput"><code>Binomial(a+b,b)</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Multinomial_theorem"; target="_top">Wikipedia</a>, <a class="ulink"
href="http://planetmath.org/MultinomialTheorem"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/MultinomialCoefficient.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-NextCombination"></a>NextCombination</span></dt><dd><pre class="synopsis">NextCombination
(v,n)</pre><p>Hämta kombination som skulle komma efter v i anrop till kombinationer, första kombination
skulle vara <strong class="userinput"><code>[1:k]</code></strong>. Denna funktion är användbar om du har
många ko
mbinationer att gå igenom och du inte vill slösa minne med att lagra dem alla.</p><p>Till exempel med
Combinations skulle du vanligen skriva en slinga som: </p><pre class="screen"><strong
class="userinput"><code>for n in Combinations (4,6) do (
EnFunktion (n)
);</code></strong>
</pre><p> Men med NextCombination skulle du skriva något som: </p><pre class="screen"><strong
class="userinput"><code>n:=[1:4];
do (
EnFunktion (n)
) while not IsNull(n:=NextCombination(n,6));</code></strong>
-</pre><p> Se även <a class="link"
href="ch11s10.html#gel-function-Combinations">Combinations</a>.</p></dd><dt><span class="term"><a
name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre class="synopsis">Pascal (i)</pre><p>Hämta Pascals
triangel som en matris. Detta kommer att returnera en (<code class="varname">i</code>+1)×(<code
class="varname">i</code>+1) nedåt diagonal matris som är Pascals triangel efter <code
class="varname">i</code> iterationer.</p><p>Se <a class="ulink" href="http://planetmath.org/PascalsTriangle";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Hämta alla permutationer av <code class="varname">k</code> tal från 1 till <code
class="varname">n</code> som en vektor av vektorer.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> eller <a cla
ss="ulink" href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alias: <code class="function">Pochhammer</code></p><p>(Pochhammer) Stigande fakultet: (n)_k =
n(n+1)…(n+(k-1)).</p><p>Se <a class="ulink" href="http://planetmath.org/RisingFactorial";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alias: <code
class="function">StirlingS1</code></p><p>Stirlingtal av första slaget.</p><p>Se <a class="ulink"
href="http://planetmath.org/StirlingNumbersOfTheFirstKind"; target="_top">Planetmath</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html"; target="_top">Mathworld</a
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Alias: <code
class="function">StirlingS2</code></p><p>Stirlingtal av andra slaget.</p><p>Se <a class="ulink"
href="http://planetmath.org/StirlingNumbersSecondKind"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Derangemang: n! gånger sum_{k=0}^n (-1)^k/k!.</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangular
(n)</pre><p>Beräkna det <code class="varname">n</code>:e triangeltalet.</p><p>Se <a class="ulink"
href="http://planetmath.org/Triangul
arNumbers" target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-nCr"></a>nCr</span></dt><dd><pre class="synopsis">nCr (n,r)</pre><p>Alias: <code
class="function">Binomial</code></p><p>Beräkna kombinationer, det vill säga binomialkoefficienten. <code
class="varname">n</code> kan vara ett godtyckligt reellt tal.</p><p>Se <a class="ulink"
href="http://planetmath.org/Choose"; target="_top">Planetmath</a> för mer information.</p></dd><dt><span
class="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre class="synopsis">nPr
(n,r)</pre><p>Beräkna antalet permutationer av storlek <code class="varname">r</code> av tal från 1 till
<code class="varname">n</code>.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> eller <a class="ulink"
href="http://en.wikipedia.org/wiki/Permutation"; target="_top">Wikipedia</a> för mer
information.</p></dd></dl></div></div><div class="
navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a
accesskey="p" href="ch11s09.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s11.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Linjär algebra </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%" align="right"
valign="top"> Kalkyl</td></tr></table></div></body></html>
+</pre><p> Se även <a class="link" href="ch11s10.html#gel-function-Combinations">Combinations</a>.</p><p>Se
<a class="ulink" href="https://en.wikipedia.org/wiki/Combination"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Pascal"></a>Pascal</span></dt><dd><pre
class="synopsis">Pascal (i)</pre><p>Hämta Pascals triangel som en matris. Detta kommer att returnera en
(<code class="varname">i</code>+1)×(<code class="varname">i</code>+1) nedåt diagonal matris som är Pascals
triangel efter <code class="varname">i</code> iterationer.</p><p>Se <a class="ulink"
href="http://planetmath.org/PascalsTriangle"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Permutations"></a>Permutations</span></dt><dd><pre class="synopsis">Permutations
(k,n)</pre><p>Hämta alla permutationer av <code class="varname">k</code> tal från 1 till <code
class="varname">n</code> som en vektor av vekt
orer.</p><p>Se <a class="ulink" href="http://mathworld.wolfram.com/Permutation.html";
target="_top">Mathworld</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Permutation";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RisingFactorial"></a>RisingFactorial</span></dt><dd><pre class="synopsis">RisingFactorial
(n,k)</pre><p>Alias: <code class="function">Pochhammer</code></p><p>(Pochhammer) Stigande fakultet: (n)_k =
n(n+1)…(n+(k-1)).</p><p>Se <a class="ulink" href="http://planetmath.org/RisingFactorial";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberFirst"></a>StirlingNumberFirst</span></dt><dd><pre
class="synopsis">StirlingNumberFirst (n,m)</pre><p>Alias: <code
class="function">StirlingS1</code></p><p>Stirlingtal av första slaget.</p><p>Se <a class="ulink"
href="http://planetmath.org/StirlingNumbersOfTheFirstKind"; target="_top">Planetma
th</a> eller <a class="ulink" href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-StirlingNumberSecond"></a>StirlingNumberSecond</span></dt><dd><pre
class="synopsis">StirlingNumberSecond (n,m)</pre><p>Alias: <code
class="function">StirlingS2</code></p><p>Stirlingtal av andra slaget.</p><p>Se <a class="ulink"
href="http://planetmath.org/StirlingNumbersSecondKind"; target="_top">Planetmath</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Subfactorial"></a>Subfactorial</span></dt><dd><pre class="synopsis">Subfactorial
(n)</pre><p>Derangemang: n! gånger sum_{k=0}^n (-1)^k/k!.</p></dd><dt><span class="term"><a
name="gel-function-Triangular"></a>Triangular</span></dt><dd><pre class="synopsis">Triangular (n)</pre>
<p>Beräkna det <code class="varname">n</code>:e triangeltalet.</p><p>Se <a class="ulink"
href="http://planetmath.org/TriangularNumbers"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-nCr"></a>nCr</span></dt><dd><pre
class="synopsis">nCr (n,r)</pre><p>Alias: <code class="function">Binomial</code></p><p>Beräkna kombinationer,
det vill säga binomialkoefficienten. <code class="varname">n</code> kan vara ett godtyckligt reellt
tal.</p><p>Se <a class="ulink" href="http://planetmath.org/Choose"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-nPr"></a>nPr</span></dt><dd><pre
class="synopsis">nPr (n,r)</pre><p>Beräkna antalet permutationer av storlek <code class="varname">r</code> av
tal från 1 till <code class="varname">n</code>.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/Permutation.html"; target="_top">Mathworld</a> eller <a class="ulink"
href="htt
ps://en.wikipedia.org/wiki/Permutation" target="_top">Wikipedia</a> för mer
information.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s09.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s11.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Linjär
algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Kalkyl</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s11.html b/help/sv/html/ch11s11.html
index f3c7201..515caa3 100644
--- a/help/sv/html/ch11s11.html
+++ b/help/sv/html/ch11s11.html
@@ -1 +1 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Kalkyl</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s10.html" title="Kombinatorik"><link rel="next" href="ch11s12.html" title="Funktioner"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Kalkyl</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s10.html">Föregående</a> </td><th width="60%"
align="center">Kapitel 11. Lista över GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s12.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-calculus"></a>Kalkyl</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integration av f med sammansatt Simpsons regel på
intervallet [a,b] med n underintervall med fel högst max(f'''')*h^4*(b-a)/180, observera att n ska vara
jämn.</p><p>Se <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> för
mer information.</p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance
(f,a,b,FjärdederivataBegränsning,Tolerans)</pre><p>Integration av f med sammansatt Simpsons regel på
intervallet [a,b] med antalet steg beräknat av fjärdederivatans begränsning och den önskade toleransen.</
p><p>Se <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Försök att beräkna derivata genom att först försöka symboliskt och sedan numeriskt.</p><p>Se
<a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Returnera en funktion som är den jämna periodiska
utvidgningen av <code class="function">f</code> med halvperiod <code class="varname">L</code>. Det vill säga
en funktion definierad på intervallet <strong class="userinput"><code>[0,L]</code></strong> utvidgad att vara
jämn på <strong class="userinput"><code>[-L,L]</code></strong> o
ch sedan utvidgad för att vara periodisk med perioden <strong
class="userinput"><code>2*L</code></strong>.</p><p>Se även <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> och <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Version 1.0.7 och
framåt.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Returnera en funktion som är en Fourierserie med
koefficienterna angivna av vektorerna <code class="varname">a</code> (sinus) och <code
class="varname">b</code> (cosinus). Observera att <strong class="userinput"><code>a@(1)</code></strong> är
den konstanta koefficienten! Det vill säga, <strong class="userinput"><code>a@(n)</code></strong> avser
termen <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, medan <strong
class="userinput"><code>b@(n)
</code></strong> avser termen <strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Antingen <code
class="varname">a</code> eller <code class="varname">b</code> kan vara <code
class="constant">null</code>.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(funk,start,ökn)</pre><p>Försök beräkna en oändlig produkt för en funktion med en
parameter.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,ökn)</pre><p>Försök beräkna en oändlig produkt för en
funktion med dubbel parameter med func(arg,n).</p></dd><dt><span class="term"
<a name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(funk,start,ökn)</pre><p>Försök beräkna en oändlig summa för en funktion med en
parameter.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre class="synopsis">InfiniteSum2
(func,arg,start,ökn)</pre><p>Försök beräkna en oändlig summa för en funktion med dubbel parameter med
func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Testa och se om en reellvärd funktion är kontinuerlig vid x0 genom att beräkna gränsvärdet
där.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Testa för differentierbarhet genom att approximera
vänster- och högergränsvärden och jämföra.</p></dd><dt><span class
="term"><a name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Beräkna vänstergränsvärdet för en reellvärd funktion vid x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Beräkna
gränsvärdet för en reellvärd funktion vid x0. Försöker beräkna både vänster- och
högergränsvärden.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Integration med mittpunktsregeln.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alias: <code
class="function">NDerivative</code></p><p>Försök beräkna numerisk derivata.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> för me
r information.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Returnera en vektor av vektorer <strong
class="userinput"><code>[a,b]</code></strong> där <code class="varname">a</code> är cosinuskoefficienterna
och <code class="varname">b</code> är sinuskoefficienterna för Fourierserien av <code
class="function">f</code> med halvperiod <code class="varname">L</code> (det vill säga definierad på <strong
class="userinput"><code>[-L,L]</code></strong> och utvidgad periodiskt) med koefficienter upp till <code
class="varname">N</code>:e deltonen beräknade numeriskt. Koefficienterna beräknas med numerisk integration
med <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_ser
ies" target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Returnera en funktion som är Fourierserien av
<code class="function">f</code> med halvperiod <code class="varname">L</code> (det vill säga definierad på
<strong class="userinput"><code>[-L,L]</code></strong> och utvidgad periodiskt) med koefficienter upp till
<code class="varname">N</code>:e deltonen beräknade numeriskt. Detta är den trigonometriska reella serien som
byggs upp av sinus och cosinus. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se
<a class="ulink" href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Returnera en vektor av
koefficienter för cosinus-Fourierserien av <code class="function">f</code> med halvperiod <code
class="varname">L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den jämna periodiska utvidgningen och beräknar
Fourierserien, som endast har cosinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a
class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>. Observera att <strong
class="userinput"><code>a@(1)</code></strong> är den konstanta koefficienten! Det vill säga, <strong
class="userinput"><code>a@(n)</code></strong> avser termen <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Returnera en funktion som är
cosinus-Fourierserien av <code class="function">f</code> med halvperiod <code class="varname">L</
code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den jämna periodiska utvidgningen och beräknar
Fourierserien, som endast har cosinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Returnera en
vektor av koefficienter för sinus-Fourierserien av <code class="function">f</code> med halvperiod <code
class="varname">L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den udda periodiska utvidgningen och beräknar
Fourierserien, som endast har sinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeriesFunc
tion</span></dt><dd><pre class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Returnera en
funktion som är sinus-Fourierserien av <code class="function">f</code> med halvperiod <code
class="varname">L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den udda periodiska utvidgningen och beräknar
Fourierserien, som endast har sinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="http://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term
"><a name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration efter regel inställd i
NumericalIntegralFunction av f från a till b med NumericalIntegralSteps steg.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Försök beräkna numerisk
vänsterderivata.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerans,upprepade_som_ger_lyckat,N)</pre><p>Försök
beräkna gränsvärdet av f(step_fun(i)) medan i går från 1 till N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Försök beräkna n
umerisk högerderivata.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Returnera en funktion som är den udda periodiska
utvidgningen av <code class="function">f</code> med halvperiod <code class="varname">L</code>. Det vill säga
en funktion definierad på intervallet <strong class="userinput"><code>[0,L]</code></strong> utvidgad att vara
udda på <strong class="userinput"><code>[-L,L]</code></strong> och sedan utvidgad för att vara periodisk med
perioden <strong class="userinput"><code>2*L</code></strong>.</p><p>Se även <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> och <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Version 1.0.7 och
framåt.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFormula</s
pan></dt><dd><pre class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Beräkna ensidig derivata med
fempunktsformel.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Beräkna ensidig derivata med
trepunktsformel.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Returnera en funktion som är den periodiska utvidgningen
av <code class="function">f</code> definierad på intervallet <strong
class="userinput"><code>[a,b]</code></strong> och har perioden <strong
class="userinput"><code>b-a</code></strong>.</p><p>Se även <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> och <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.</p><p>V
ersion 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre class="synopsis">RightLimit
(f,x0)</pre><p>Beräkna högergränsvärdet för en reellvärd funktion vid x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Beräkna tvåsidig derivata med
fempunktsformel.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Beräkna tvåsidig derivata med
trepunktsformel.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Kalkyl</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
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width="20%" align="left"><a accesskey="p" href="ch11s10.html">Föregående</a> </td><th width="60%"
align="center">Kapitel 11. Lista över GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s12.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-calculus"></a>Kalkyl</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRule"></a>CompositeSimpsonsRule</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRule (f,a,b,n)</pre><p>Integration av f med sammansatt Simpsons regel på
intervallet [a,b] med n underintervall med fel högst max(f'''')*h^4*(b-a)/180, observera att n ska vara
jämn.</p><p>Se <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> för
mer information.</p></dd><dt><span class="term"><a
name="gel-function-CompositeSimpsonsRuleTolerance"></a>CompositeSimpsonsRuleTolerance</span></dt><dd><pre
class="synopsis">CompositeSimpsonsRuleTolerance
(f,a,b,FjärdederivataBegränsning,Tolerans)</pre><p>Integration av f med sammansatt Simpsons regel på
intervallet [a,b] med antalet steg beräknat av fjärdederivatans begränsning och den önskade toleransen.</
p><p>Se <a class="ulink" href="http://planetmath.org/SimpsonsRule"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-Derivative"></a>Derivative</span></dt><dd><pre class="synopsis">Derivative
(f,x0)</pre><p>Försök att beräkna derivata genom att först försöka symboliskt och sedan numeriskt.</p><p>Se
<a class="ulink" href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-EvenPeriodicExtension"></a>EvenPeriodicExtension</span></dt><dd><pre
class="synopsis">EvenPeriodicExtension (f,L)</pre><p>Returnera en funktion som är den jämna periodiska
utvidgningen av <code class="function">f</code> med halvperiod <code class="varname">L</code>. Det vill säga
en funktion definierad på intervallet <strong class="userinput"><code>[0,L]</code></strong> utvidgad att vara
jämn på <strong class="userinput"><code>[-L,L]</code></strong> o
ch sedan utvidgad för att vara periodisk med perioden <strong
class="userinput"><code>2*L</code></strong>.</p><p>Se även <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> och <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Version 1.0.7 och
framåt.</p></dd><dt><span class="term"><a
name="gel-function-FourierSeriesFunction"></a>FourierSeriesFunction</span></dt><dd><pre
class="synopsis">FourierSeriesFunction (a,b,L)</pre><p>Returnera en funktion som är en Fourierserie med
koefficienterna angivna av vektorerna <code class="varname">a</code> (sinus) och <code
class="varname">b</code> (cosinus). Observera att <strong class="userinput"><code>a@(1)</code></strong> är
den konstanta koefficienten! Det vill säga, <strong class="userinput"><code>a@(n)</code></strong> avser
termen <strong class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>, medan <strong
class="userinput"><code>b@(n)
</code></strong> avser termen <strong class="userinput"><code>sin(x*n*pi/L)</code></strong>. Antingen <code
class="varname">a</code> eller <code class="varname">b</code> kan vara <code
class="constant">null</code>.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct"></a>InfiniteProduct</span></dt><dd><pre class="synopsis">InfiniteProduct
(funk,start,ökn)</pre><p>Försök beräkna en oändlig produkt för en funktion med en
parameter.</p></dd><dt><span class="term"><a
name="gel-function-InfiniteProduct2"></a>InfiniteProduct2</span></dt><dd><pre
class="synopsis">InfiniteProduct2 (func,arg,start,ökn)</pre><p>Försök beräkna en oändlig produkt för en
funktion med dubbel parameter med func(arg,n).</p></dd><dt><span class="term
"><a name="gel-function-InfiniteSum"></a>InfiniteSum</span></dt><dd><pre class="synopsis">InfiniteSum
(funk,start,ökn)</pre><p>Försök beräkna en oändlig summa för en funktion med en parameter.</p></dd><dt><span
class="term"><a name="gel-function-InfiniteSum2"></a>InfiniteSum2</span></dt><dd><pre
class="synopsis">InfiniteSum2 (func,arg,start,ökn)</pre><p>Försök beräkna en oändlig summa för en funktion
med dubbel parameter med func(arg,n).</p></dd><dt><span class="term"><a
name="gel-function-IsContinuous"></a>IsContinuous</span></dt><dd><pre class="synopsis">IsContinuous
(f,x0)</pre><p>Testa och se om en reellvärd funktion är kontinuerlig vid x0 genom att beräkna gränsvärdet
där.</p></dd><dt><span class="term"><a
name="gel-function-IsDifferentiable"></a>IsDifferentiable</span></dt><dd><pre
class="synopsis">IsDifferentiable (f,x0)</pre><p>Testa för differentierbarhet genom att approximera vänster-
och högergränsvärden och jämföra.</p></dd><dt><span clas
s="term"><a name="gel-function-LeftLimit"></a>LeftLimit</span></dt><dd><pre class="synopsis">LeftLimit
(f,x0)</pre><p>Beräkna vänstergränsvärdet för en reellvärd funktion vid x0.</p></dd><dt><span class="term"><a
name="gel-function-Limit"></a>Limit</span></dt><dd><pre class="synopsis">Limit (f,x0)</pre><p>Beräkna
gränsvärdet för en reellvärd funktion vid x0. Försöker beräkna både vänster- och
högergränsvärden.</p></dd><dt><span class="term"><a
name="gel-function-MidpointRule"></a>MidpointRule</span></dt><dd><pre class="synopsis">MidpointRule
(f,a,b,n)</pre><p>Integration med mittpunktsregeln.</p></dd><dt><span class="term"><a
name="gel-function-NumericalDerivative"></a>NumericalDerivative</span></dt><dd><pre
class="synopsis">NumericalDerivative (f,x0)</pre><p>Alias: <code
class="function">NDerivative</code></p><p>Försök beräkna numerisk derivata.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Derivative"; target="_top">Wikipedia</a> för m
er information.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesCoefficients"></a>NumericalFourierSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesCoefficients (f,L,N)</pre><p>Returnera en vektor av vektorer <strong
class="userinput"><code>[a,b]</code></strong> där <code class="varname">a</code> är cosinuskoefficienterna
och <code class="varname">b</code> är sinuskoefficienterna för Fourierserien av <code
class="function">f</code> med halvperiod <code class="varname">L</code> (det vill säga definierad på <strong
class="userinput"><code>[-L,L]</code></strong> och utvidgad periodiskt) med koefficienter upp till <code
class="varname">N</code>:e deltonen beräknade numeriskt. Koefficienterna beräknas med numerisk integration
med <a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fourier_s
eries" target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSeriesFunction"></a>NumericalFourierSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierSeriesFunction (f,L,N)</pre><p>Returnera en funktion som är Fourierserien av
<code class="function">f</code> med halvperiod <code class="varname">L</code> (det vill säga definierad på
<strong class="userinput"><code>[-L,L]</code></strong> och utvidgad periodiskt) med koefficienter upp till
<code class="varname">N</code>:e deltonen beräknade numeriskt. Detta är den trigonometriska reella serien som
byggs upp av sinus och cosinus. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>S
e <a class="ulink" href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://mathworld.wolfram.com/FourierSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesCoefficients"></a>NumericalFourierCosineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesCoefficients (f,L,N)</pre><p>Returnera en vektor av
koefficienter för cosinus-Fourierserien av <code class="function">f</code> med halvperiod <code
class="varname">L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den jämna periodiska utvidgningen och beräknar
Fourierserien, som endast har cosinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med
<a class="link" href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>. Observera att <strong
class="userinput"><code>a@(1)</code></strong> är den konstanta koefficienten! Det vill säga, <strong
class="userinput"><code>a@(n)</code></strong> avser termen <strong
class="userinput"><code>cos(x*(n-1)*pi/L)</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierCosineSeriesFunction"></a>NumericalFourierCosineSeriesFunction</span></dt><dd><pre
class="synopsis">NumericalFourierCosineSeriesFunction (f,L,N)</pre><p>Returnera en funktion som är
cosinus-Fourierserien av <code class="function">f</code> med halvperiod <code class="varname"
L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den jämna periodiska utvidgningen och beräknar
Fourierserien, som endast har cosinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierCosineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesCoefficients"></a>NumericalFourierSineSeriesCoefficients</span></dt><dd><pre
class="synopsis">NumericalFourierSineSeriesCoefficients (f,L,N)</pre><p>Returner
a en vektor av koefficienter för sinus-Fourierserien av <code class="function">f</code> med halvperiod <code
class="varname">L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den udda periodiska utvidgningen och beräknar
Fourierserien, som endast har sinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-NumericalFourierSineSeriesFunction"></a>NumericalFourierSineSeri
esFunction</span></dt><dd><pre class="synopsis">NumericalFourierSineSeriesFunction (f,L,N)</pre><p>Returnera
en funktion som är sinus-Fourierserien av <code class="function">f</code> med halvperiod <code
class="varname">L</code>. Det vill säga vi tar <code class="function">f</code> definierad på <strong
class="userinput"><code>[0,L]</code></strong> och tar den udda periodiska utvidgningen och beräknar
Fourierserien, som endast har sinustermer. Serien beräknas upp till <code class="varname">N</code>:e
deltonen. Koefficienterna beräknas med numerisk integration med <a class="link"
href="ch11s11.html#gel-function-NumericalIntegral"><code
class="function">NumericalIntegral</code></a>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Fourier_series"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/FourierSineSeries.html"; target="_top">Mathworld</a> för mer
information.</p><p>Version 1.0.7 och framåt.</p></dd><dt><span clas
s="term"><a name="gel-function-NumericalIntegral"></a>NumericalIntegral</span></dt><dd><pre
class="synopsis">NumericalIntegral (f,a,b)</pre><p>Integration efter regel inställd i
NumericalIntegralFunction av f från a till b med NumericalIntegralSteps steg.</p></dd><dt><span
class="term"><a name="gel-function-NumericalLeftDerivative"></a>NumericalLeftDerivative</span></dt><dd><pre
class="synopsis">NumericalLeftDerivative (f,x0)</pre><p>Försök beräkna numerisk
vänsterderivata.</p></dd><dt><span class="term"><a
name="gel-function-NumericalLimitAtInfinity"></a>NumericalLimitAtInfinity</span></dt><dd><pre
class="synopsis">NumericalLimitAtInfinity (_f,step_fun,tolerans,upprepade_som_ger_lyckat,N)</pre><p>Försök
beräkna gränsvärdet av f(step_fun(i)) medan i går från 1 till N.</p></dd><dt><span class="term"><a
name="gel-function-NumericalRightDerivative"></a>NumericalRightDerivative</span></dt><dd><pre
class="synopsis">NumericalRightDerivative (f,x0)</pre><p>Försök ber
äkna numerisk högerderivata.</p></dd><dt><span class="term"><a
name="gel-function-OddPeriodicExtension"></a>OddPeriodicExtension</span></dt><dd><pre
class="synopsis">OddPeriodicExtension (f,L)</pre><p>Returnera en funktion som är den udda periodiska
utvidgningen av <code class="function">f</code> med halvperiod <code class="varname">L</code>. Det vill säga
en funktion definierad på intervallet <strong class="userinput"><code>[0,L]</code></strong> utvidgad att vara
udda på <strong class="userinput"><code>[-L,L]</code></strong> och sedan utvidgad för att vara periodisk med
perioden <strong class="userinput"><code>2*L</code></strong>.</p><p>Se även <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a> och <a class="link"
href="ch11s11.html#gel-function-PeriodicExtension">PeriodicExtension</a>.</p><p>Version 1.0.7 och
framåt.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedFivePointFormula"></a>OneSidedFivePointFor
mula</span></dt><dd><pre class="synopsis">OneSidedFivePointFormula (f,x0,h)</pre><p>Beräkna ensidig derivata
med fempunktsformel.</p></dd><dt><span class="term"><a
name="gel-function-OneSidedThreePointFormula"></a>OneSidedThreePointFormula</span></dt><dd><pre
class="synopsis">OneSidedThreePointFormula (f,x0,h)</pre><p>Beräkna ensidig derivata med
trepunktsformel.</p></dd><dt><span class="term"><a
name="gel-function-PeriodicExtension"></a>PeriodicExtension</span></dt><dd><pre
class="synopsis">PeriodicExtension (f,a,b)</pre><p>Returnera en funktion som är den periodiska utvidgningen
av <code class="function">f</code> definierad på intervallet <strong
class="userinput"><code>[a,b]</code></strong> och har perioden <strong
class="userinput"><code>b-a</code></strong>.</p><p>Se även <a class="link"
href="ch11s11.html#gel-function-OddPeriodicExtension">OddPeriodicExtension</a> och <a class="link"
href="ch11s11.html#gel-function-EvenPeriodicExtension">EvenPeriodicExtension</a>.<
/p><p>Version 1.0.7 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-RightLimit"></a>RightLimit</span></dt><dd><pre class="synopsis">RightLimit
(f,x0)</pre><p>Beräkna högergränsvärdet för en reellvärd funktion vid x0.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedFivePointFormula"></a>TwoSidedFivePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedFivePointFormula (f,x0,h)</pre><p>Beräkna tvåsidig derivata med
fempunktsformel.</p></dd><dt><span class="term"><a
name="gel-function-TwoSidedThreePointFormula"></a>TwoSidedThreePointFormula</span></dt><dd><pre
class="synopsis">TwoSidedThreePointFormula (f,x0,h)</pre><p>Beräkna tvåsidig derivata med
trepunktsformel.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
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class="titlepage"><div><div><h2 class="title" styl
e="clear: both"><a name="genius-gel-function-list-functions"></a>Funktioner</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Alias:
<code class="function">Arg</code><code class="function">arg</code></p><p>argument (vinkel) för komplext
tal.</p></dd><dt><span class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre
class="synopsis">BesselJ0 (x)</pre><p>Besselfunktion av första slaget av ordning 0. Endast implementerad för
reella tal.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1
(x)</pre><p>Besselfunktion av första slaget av ordning 1. Endast implementerad för reella t
al.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn
(n,x)</pre><p>Besselfunktion av första slaget av ordning <code class="varname">n</code>. Endast implementerad
för reella tal.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0
(x)</pre><p>Besselfunktion av andra slaget av ordning 0. Endast implementerad för reella tal.</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> för mer
information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span class="term
"><a name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1
(x)</pre><p>Besselfunktion av andra slaget av ordning 1. Endast implementerad för reella tal.</p><p>Se <a
class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> för mer
information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn
(n,x)</pre><p>Besselfunktion av andra slaget av ordning <code class="varname">n</code>. Endast implementerad
för reella tal.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre
class="synopsis">DirichletKernel (n,t)</pre><p>Dirichletkärna av ordning <code
class="varname">n</code>.</p></dd><dt
<span class="term"><a name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre
class="synopsis">DiscreteDelta (v)</pre><p>Returnerar 1 om och endast om alla element är
noll.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alias: <code class="function">erf</code></p><p>Felfunktionen, 2/sqrt(2) * int_0^x e^(-t^2)
dt.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Error_function";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/ErrorFunction";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-FejerKernel"></a>FejerKernel</span></dt><dd><pre class="synopsis">FejerKernel
(n,t)</pre><p>Fejerkärna av ordning <code class="varname">n</code> beräknad vid <code
class="varname">t</code></p><p>Se <a class="ulink" href="http://planetmath.org/FejerKernel";
target="_top">Planet
math</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alias: <code class="function">Gamma</code></p><p>Gammafunktionen. För närvarande bara
implementerad för reella värden.</p><p>Se <a class="ulink" href="http://planetmath.org/GammaFunction";
target="_top">Planetmath</a> eller <a class="ulink" href="http://en.wikipedia.org/wiki/Gamma_function";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returnerar 1 om och endast om alla element är lika.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW
(x)</pre><p>Huvudgrenen av Lamberts W-funktion beräknad endast för reella värden större än eller lika med
<strong class="userinput">
<code>-1/e</code></strong>. Det vill säga <code class="function">LambertW</code> är inversen av <strong
class="userinput"><code>x*e^x</code></strong>. Även för reella värden på <code class="varname">x</code> är
detta uttryck inte 1 till 1 och har därför två grenar över <strong
class="userinput"><code>[-1/e,0)</code></strong>. Se <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> för den andra reella
grenen.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1
(x)</pre><p>Minus-ett-grenen av Lamberts W-funktion beräknad endast för reella värden större än eller lika
med <strong class="userinput"><code>-1/e</code></strong> och mindre än 0. Det vill säga <code class="func
tion">LambertWm1</code> är den andra grenen av inversen av <strong
class="userinput"><code>x*e^x</code></strong>. Se <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> för
huvudgrenen.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (funk,x,ökn)</pre><p>Hitta det första värdet där f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Möbiusavbildning av skivan till sig själv som avbildar a
till 0.</p><p>Se <a class="ulink" href="http://planetmath.org/MobiusTransformation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a name="gel-function-M
oebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Möbiusavbildning som använder dubbelförhållandet som tar z2,z3,z4 till 1, 0 respektive
oändligheten.</p><p>Se <a class="ulink" href="http://planetmath.org/MobiusTransformation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Möbiusavbildning som använder
dubbelförhållandet som tar oändligheten till oändligheten och z2,z3 till 1 respektive 0.</p><p>Se <a
class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Möbiusavbildni
ng som använder dubbelförhållandet som tar oändligheten till 1 och z3,z4 till 0 respektive
oändligheten.</p><p>Se <a class="ulink" href="http://planetmath.org/MobiusTransformation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Möbiusavbildning som använder dubbelförhållandet
som tar oändligheten till 0 och z2,z4 till 1 respektive oändligheten.</p><p>Se <a class="ulink"
href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poissonkärna på D(0,1) (inte normaliserad till 1, det vill säga integral av detta är
2pi).</p></dd><dt><span class="term"><a name="gel-function-Poisso
nKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre class="synopsis">PoissonKernelRadius
(r,sigma)</pre><p>Poissonkärna på D(0,R) (inte normaliserad till 1).</p></dd><dt><span class="term"><a
name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre class="synopsis">RiemannZeta
(x)</pre><p>Alias: <code class="function">zeta</code></p><p>Riemanns zetafunktion. För närvarande bara
implementerad för reella värden.</p><p>Se <a class="ulink" href="http://planetmath.org/RiemannZetaFunction";
target="_top">Planetmath</a> eller <a class="ulink" href="http://en.wikipedia.org/wiki/Riemann_zeta_function";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre class="synopsis">UnitStep
(x)</pre><p>Enhetsstegfunktionen är 0 för x<0, 1 annars. Detta är integralen för Diracs delta-funktion.
Också kallad Heavisidefunktionen.</p><p>Se <a class="ulink" href="http://en.wikipe
dia.org/wiki/Unit_step" target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-cis"></a>cis</span></dt><dd><pre class="synopsis">cis (x)</pre><p><code
class="function">cis</code>-funktionen, detta är samma sak som <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Konvertera
grader till radianer.</p></dd><dt><span class="term"><a
name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg (x)</pre><p>Konvertera
radianer till grader.</p></dd><dt><span class="term"><a name="gel-function-sinc"></a>sinc</span></dt><dd><pre
class="synopsis">sinc (x)</pre><p>Beräknar den onormaliserade sinc-funktionen, det vill säga <strong
class="userinput"><code>sin(x)/x</code></strong>. Om du vill ha den normaliserade funktionen, anropa <strong
class="userinput"><code>sinc(pi*x
)</code></strong>.</p><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Sinc";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och
framåt.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s11.html">Föregående</a> </td><td
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</td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
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href="ch11s13.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" styl
e="clear: both"><a name="genius-gel-function-list-functions"></a>Funktioner</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Argument"></a>Argument</span></dt><dd><pre class="synopsis">Argument (z)</pre><p>Alias:
<code class="function">Arg</code><code class="function">arg</code></p><p>argument (vinkel) för komplext
tal.</p></dd><dt><span class="term"><a name="gel-function-BesselJ0"></a>BesselJ0</span></dt><dd><pre
class="synopsis">BesselJ0 (x)</pre><p>Besselfunktion av första slaget av ordning 0. Endast implementerad för
reella tal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-BesselJ1"></a>BesselJ1</span></dt><dd><pre class="synopsis">BesselJ1
(x)</pre><p>Besselfunktion av första slaget av ordning 1. Endast implementerad för reella
tal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-BesselJn"></a>BesselJn</span></dt><dd><pre class="synopsis">BesselJn
(n,x)</pre><p>Besselfunktion av första slaget av ordning <code class="varname">n</code>. Endast implementerad
för reella tal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-BesselY0"></a>BesselY0</span></dt><dd><pre class="synopsis">BesselY0
(x)</pre><p>Besselfunktion av andra slaget av ordning 0. Endast implementerad för reella tal.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> för mer
information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span class="
term"><a name="gel-function-BesselY1"></a>BesselY1</span></dt><dd><pre class="synopsis">BesselY1
(x)</pre><p>Besselfunktion av andra slaget av ordning 1. Endast implementerad för reella tal.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions"; target="_top">Wikipedia</a> för mer
information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-BesselYn"></a>BesselYn</span></dt><dd><pre class="synopsis">BesselYn
(n,x)</pre><p>Besselfunktion av andra slaget av ordning <code class="varname">n</code>. Endast implementerad
för reella tal.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Bessel_functions";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.16 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-DirichletKernel"></a>DirichletKernel</span></dt><dd><pre
class="synopsis">DirichletKernel (n,t)</pre><p>Dirichletkärna av ordning <code class="varname">n</code>.</p></
dd><dt><span class="term"><a name="gel-function-DiscreteDelta"></a>DiscreteDelta</span></dt><dd><pre
class="synopsis">DiscreteDelta (v)</pre><p>Returnerar 1 om och endast om alla element är
noll.</p></dd><dt><span class="term"><a
name="gel-function-ErrorFunction"></a>ErrorFunction</span></dt><dd><pre class="synopsis">ErrorFunction
(x)</pre><p>Alias: <code class="function">erf</code></p><p>Felfunktionen, 2/sqrt(2) * int_0^x e^(-t^2)
dt.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Error_function"; target="_top">Wikipedia</a>
eller <a class="ulink" href="http://planetmath.org/ErrorFunction"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-FejerKernel"></a>FejerKernel</span></dt><dd><pre class="synopsis">FejerKernel
(n,t)</pre><p>Fejerkärna av ordning <code class="varname">n</code> beräknad vid <code
class="varname">t</code></p><p>Se <a class="ulink" href="http://planetmath.org/FejerKernel"; target="_top">
Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-GammaFunction"></a>GammaFunction</span></dt><dd><pre class="synopsis">GammaFunction
(x)</pre><p>Alias: <code class="function">Gamma</code></p><p>Gammafunktionen. För närvarande bara
implementerad för reella värden.</p><p>Se <a class="ulink" href="http://planetmath.org/GammaFunction";
target="_top">Planetmath</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Gamma_function";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-KroneckerDelta"></a>KroneckerDelta</span></dt><dd><pre class="synopsis">KroneckerDelta
(v)</pre><p>Returnerar 1 om och endast om alla element är lika.</p></dd><dt><span class="term"><a
name="gel-function-LambertW"></a>LambertW</span></dt><dd><pre class="synopsis">LambertW
(x)</pre><p>Huvudgrenen av Lamberts W-funktion beräknad endast för reella värden större än eller lika med
<strong class="user
input"><code>-1/e</code></strong>. Det vill säga <code class="function">LambertW</code> är inversen av
<strong class="userinput"><code>x*e^x</code></strong>. Även för reella värden på <code
class="varname">x</code> är detta uttryck inte 1 till 1 och har därför två grenar över <strong
class="userinput"><code>[-1/e,0)</code></strong>. Se <a class="link"
href="ch11s12.html#gel-function-LambertWm1"><code class="function">LambertWm1</code></a> för den andra reella
grenen.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-LambertWm1"></a>LambertWm1</span></dt><dd><pre class="synopsis">LambertWm1
(x)</pre><p>Minus-ett-grenen av Lamberts W-funktion beräknad endast för reella värden större än eller lika
med <strong class="userinput"><code>-1/e</code></strong> och mindre än 0. Det vill säga <code cla
ss="function">LambertWm1</code> är den andra grenen av inversen av <strong
class="userinput"><code>x*e^x</code></strong>. Se <a class="link"
href="ch11s12.html#gel-function-LambertW"><code class="function">LambertW</code></a> för
huvudgrenen.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Lambert_W_function";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MinimizeFunction"></a>MinimizeFunction</span></dt><dd><pre
class="synopsis">MinimizeFunction (funk,x,ökn)</pre><p>Hitta det första värdet där f(x)=0.</p></dd><dt><span
class="term"><a name="gel-function-MoebiusDiskMapping"></a>MoebiusDiskMapping</span></dt><dd><pre
class="synopsis">MoebiusDiskMapping (a,z)</pre><p>Möbiusavbildning av skivan till sig själv som avbildar a
till 0.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/Mob
iusTransformation" target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMapping"></a>MoebiusMapping</span></dt><dd><pre class="synopsis">MoebiusMapping
(z,z2,z3,z4)</pre><p>Möbiusavbildning som använder dubbelförhållandet som tar z2,z3,z4 till 1, 0 respektive
oändligheten.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation";
target="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/MobiusTransformation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToInfty"></a>MoebiusMappingInftyToInfty</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToInfty (z,z2,z3)</pre><p>Möbiusavbildning som använder
dubbelförhållandet som tar oändligheten till oändligheten och z2,z3 till 1 respektive 0.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation"; t
arget="_top">Wikipedia</a> eller <a class="ulink" href="http://planetmath.org/MobiusTransformation";
target="_top">Planetmath</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToOne"></a>MoebiusMappingInftyToOne</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToOne (z,z3,z4)</pre><p>Möbiusavbildning som använder dubbelförhållandet
som tar oändligheten till 1 och z3,z4 till 0 respektive oändligheten.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-MoebiusMappingInftyToZero"></a>MoebiusMappingInftyToZero</span></dt><dd><pre
class="synopsis">MoebiusMappingInftyToZero (z,z2,z4)</pre><p>Möbiusavbildning som använder dubbelförhållandet
som tar oändligheten till 0 och z2,z4
till 1 respektive oändligheten.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation"; target="_top">Wikipedia</a> eller <a
class="ulink" href="http://planetmath.org/MobiusTransformation"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernel"></a>PoissonKernel</span></dt><dd><pre class="synopsis">PoissonKernel
(r,sigma)</pre><p>Poissonkärna på D(0,1) (inte normaliserad till 1, det vill säga integral av detta är
2pi).</p></dd><dt><span class="term"><a
name="gel-function-PoissonKernelRadius"></a>PoissonKernelRadius</span></dt><dd><pre
class="synopsis">PoissonKernelRadius (r,sigma)</pre><p>Poissonkärna på D(0,R) (inte normaliserad till
1).</p></dd><dt><span class="term"><a name="gel-function-RiemannZeta"></a>RiemannZeta</span></dt><dd><pre
class="synopsis">RiemannZeta (x)</pre><p>Alias: <code class="function">zeta</code></p><p>Riemanns
zetafunktion. För närvarande bara im
plementerad för reella värden.</p><p>Se <a class="ulink" href="http://planetmath.org/RiemannZetaFunction";
target="_top">Planetmath</a> eller <a class="ulink"
href="https://en.wikipedia.org/wiki/Riemann_zeta_function"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-UnitStep"></a>UnitStep</span></dt><dd><pre
class="synopsis">UnitStep (x)</pre><p>Enhetsstegfunktionen är 0 för x<0, 1 annars. Detta är integralen för
Diracs delta-funktion. Också kallad Heavisidefunktionen.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Unit_step"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-cis"></a>cis</span></dt><dd><pre
class="synopsis">cis (x)</pre><p><code class="function">cis</code>-funktionen, detta är samma sak som <strong
class="userinput"><code>cos(x)+1i*sin(x)</code></strong></p></dd><dt><span class="term"><a
name="gel-function-deg2rad"></a>deg2rad</
span></dt><dd><pre class="synopsis">deg2rad (x)</pre><p>Konvertera grader till radianer.</p></dd><dt><span
class="term"><a name="gel-function-rad2deg"></a>rad2deg</span></dt><dd><pre class="synopsis">rad2deg
(x)</pre><p>Konvertera radianer till grader.</p></dd><dt><span class="term"><a
name="gel-function-sinc"></a>sinc</span></dt><dd><pre class="synopsis">sinc (x)</pre><p>Beräknar den
onormaliserade sinc-funktionen, det vill säga <strong class="userinput"><code>sin(x)/x</code></strong>. Om du
vill ha den normaliserade funktionen, anropa <strong
class="userinput"><code>sinc(pi*x)</code></strong>.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Sinc"; target="_top">Wikipedia</a> för mer information.</p><p>Version
1.0.16 och framåt.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s11.html">Föregående</a> </td><td width="20%" align="center"><a acces
skey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
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diff --git a/help/sv/html/ch11s13.html b/help/sv/html/ch11s13.html
index 185629a..1ec0189 100644
--- a/help/sv/html/ch11s13.html
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-<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Ekvationslösning</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s12.html" title="Funktioner"><link
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accesskey="p" href="ch11s12.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
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href="ch11s14.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="t
itle" style="clear: both"><a
name="genius-gel-function-list-equation-solving"></a>Ekvationslösning</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre class="synopsis">CubicFormula
(p)</pre><p>Beräkna rötter för ett tredjegradspolynom med formel. Polynomet ska anges som en vektor av
koefficienter. Det vill säga <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> motsvarar
vektorn <strong class="userinput"><code>[1,2,0,4]</code></strong>. Returnerar en kolumnvektor av de tre
lösningarna. Den första lösningen är alltid den reella eftersom ett tredjegradspolynom alltid har en reell
lösning.</p><p>Se <a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>, <a
class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html"; target="_top">Mathworld</a> eller <a
class="ulink" href="http://en.wikipedia.org/wiki/
Cubic_equation" target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>Använd Eulers klassiska metod för att numeriskt lösa y'=f(x,y) för initialt <code
class="varname">x0</code>, <code class="varname">y0</code> som går till <code class="varname">x1</code> med
<code class="varname">n</code> inkrement, returnerar <code class="varname">y</code> vid <code
class="varname">x1</code>. Om du inte explicit vill använda Eulers metod bör du verkligen överväga att
använda <a class="link" href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a> för lösning av
ODE.</p><p>System kan lösas genom att helt enkelt låta <code class="varname">y</code> vara en (kolumn)vektor
överallt. Det vill säga, <code class="varname">y0</code> kan vara en vektor i vilket fall <code
class="varname">f</code> bör ta ett tal <code class="varnam
e">x</code> och en vektor av samma storlek som det andra argumentet och bör returnera en vektor av samma
storlek.</p><p>Se <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> eller <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>Använd Eulers klassiska metod för att numeriskt lösa
y'=f(x,y) för initialt <code class="varname">x0</code>, <code class="varname">y0</code> som går till <code
class="varname">x1</code> med <code class="varname">n</code> inkrement, returnerar en 2×(<strong
class="userinput"><code>n+1</code></strong>)-matris med <code class="varname">x</code>- och <code
class="varname">y</code>-värdena. Om du inte explicit vill använda Eulers metod bör du verkligen överv�
�ga att använda <a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a> för
lösning av ODE. Lämplig för att koppla ihop med <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> eller <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+<html><head><meta http-equiv="Content-Type" content="text/html;
charset=UTF-8"><title>Ekvationslösning</title><meta name="generator" content="DocBook XSL Stylesheets
V1.79.1"><link rel="home" href="index.html" title="Handbok för Genius"><link rel="up" href="ch11.html"
title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev" href="ch11s12.html" title="Funktioner"><link
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header"><tr><th colspan="3" align="center">Ekvationslösning</th></tr><tr><td width="20%" align="left"><a
accesskey="p" href="ch11s12.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s14.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="t
itle" style="clear: both"><a
name="genius-gel-function-list-equation-solving"></a>Ekvationslösning</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre class="synopsis">CubicFormula
(p)</pre><p>Beräkna rötter för ett tredjegradspolynom med formel. Polynomet ska anges som en vektor av
koefficienter. Det vill säga <strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> motsvarar
vektorn <strong class="userinput"><code>[1,2,0,4]</code></strong>. Returnerar en kolumnvektor av de tre
lösningarna. Den första lösningen är alltid den reella eftersom ett tredjegradspolynom alltid har en reell
lösning.</p><p>Se <a class="ulink" href="http://planetmath.org/CubicFormula"; target="_top">Planetmath</a>, <a
class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html"; target="_top">Mathworld</a> eller <a
class="ulink" href="https://en.wikipedia.org/wiki
/Cubic_equation" target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre class="synopsis">EulersMethod
(f,x0,y0,x1,n)</pre><p>Använd Eulers klassiska metod för att numeriskt lösa y'=f(x,y) för initialt <code
class="varname">x0</code>, <code class="varname">y0</code> som går till <code class="varname">x1</code> med
<code class="varname">n</code> inkrement, returnerar <code class="varname">y</code> vid <code
class="varname">x1</code>. Om du inte explicit vill använda Eulers metod bör du verkligen överväga att
använda <a class="link" href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a> för lösning av
ODE.</p><p>System kan lösas genom att helt enkelt låta <code class="varname">y</code> vara en (kolumn)vektor
överallt. Det vill säga, <code class="varname">y0</code> kan vara en vektor i vilket fall <code
class="varname">f</code> bör ta ett tal <code class="varna
me">x</code> och en vektor av samma storlek som det andra argumentet och bör returnera en vektor av samma
storlek.</p><p>Se <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre
class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p>Använd Eulers klassiska metod för att numeriskt lösa
y'=f(x,y) för initialt <code class="varname">x0</code>, <code class="varname">y0</code> som går till <code
class="varname">x1</code> med <code class="varname">n</code> inkrement, returnerar en (<strong
class="userinput"><code>n+1</code></strong>)×2-matris med <code class="varname">x</code>- och <code
class="varname">y</code>-värdena. Om du inte explicit vill använda Eulers metod bör du verkligen över
väga att använda <a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a> för
lösning av ODE. Lämplig för att koppla ihop med <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> eller <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line =
EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponentiell
tillväxt");</code></strong>
</pre><p>System kan lösas genom att helt enkelt låta <code class="varname">y</code> vara en (kolumn)vektor
överallt. Det vill säga, <code class="varname">y0</code> kan vara en vektor i vilket fall <code
class="varname">f</code> bör ta ett tal <code class="varname">x</code> och en vektor av samma storlek som det
andra argumentet och bör returnera en vektor av samma storlek.</p><p>Utdata för ett system är fortfarande en
n×2-matris där den andra posten är en vektor. Om du vill rita linjen, se till att använda radvektorer och
platta sedan till matrisen med <a class="link" href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>
och välj de rätta kolumnerna. Exempel: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
@@ -9,9 +9,9 @@
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","Första");</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Andra");</code></strong>
-</pre><p>Se <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> eller <a class="ulink" href="http://en.wikipedia.org/wiki/Eulers_method";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.10 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Hitta rot för en funktion med bisektionsmetoden.
<code class="varname">a</code> och <code class="varname">b</code> är det ursprungliga gissningsintervallet,
<strong class="userinput"><code>f(a)</code></strong> och <strong class="userinput"><code>f(b)</code></strong>
måste ha olika tecken. <code class="varname">TOL</code> är den önskade toleransen och <code
class="varname">N</code> är gränsen för hur många iterationer att köra, 0 betyder ingen gräns. Funktionen
returnerar en vektor <strong class="userinput"><code>[lyckad,värde,
iteration]</code></strong>, där <code class="varname">lyckad</code> är ett booleskt värde som indikerar om
den lyckats, <code class="varname">värde</code> är det sista beräknade värdet, och <code
class="varname">iteration</code> är antalet utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Hitta rot för en funktion med regula
falsi-metoden. <code class="varname">a</code> och <code class="varname">b</code> är det ursprungliga
gissningsintervallet, <strong class="userinput"><code>f(a)</code></strong> och <strong
class="userinput"><code>f(b)</code></strong> måste ha olika tecken. <code class="varname">TOL</code> är den
önskade toleransen och <code class="varname">N</code> är gränsen för hur många iterationer att köra, 0
betyder ingen gräns. Funktionen returnerar en vektor <strong class="userinput"><code>[lyc
kad,värde,iteration]</code></strong>, där <code class="varname">lyckad</code> är ett booleskt värde som
indikerar om den lyckats, <code class="varname">värde</code> är det sista beräknade värdet, och <code
class="varname">iteration</code> är antalet utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Hitta rot för en funktion med Mullers
metod. <code class="varname">TOL</code> är den önskade toleransen och <code class="varname">N</code> är
gränsen för antal iterationer att köra, 0 betyder ingen gräns. Funktionen returnerar en vektor <strong
class="userinput"><code>[lyckad,värde,iteration]</code></strong>, där <code class="varname">lyckad</code> är
ett booleskt värde som indikerar om den lyckats, <code class="varname">värde</code> är det sista beräknade
värdet, och <code class="varname">iteration
</code> är antalet utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Hitta rot för en funktion med sekantmetoden. <code class="varname">a</code> och <code
class="varname">b</code> är det ursprungliga gissningsintervallet, <strong
class="userinput"><code>f(a)</code></strong> och <strong class="userinput"><code>f(b)</code></strong> måste
ha olika tecken. <code class="varname">TOL</code> är den önskade toleransen och <code
class="varname">N</code> är gränsen för hur många iterationer att köra, 0 betyder ingen gräns. Funktionen
returnerar en vektor <strong class="userinput"><code>[lyckad,värde,iteration]</code></strong>, där <code
class="varname">lyckad</code> är ett booleskt värde som indikerar om den lyckats, <code
class="varname">värde</code> är det sista beräknade värdet, och <code class="varname">iteration</code> är
antal
et utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,gissning,epsilon,maxn)</pre><p>Hitta nollpunkter med Halleys metod. <code class="varname">f</code>
är funktionen, <code class="varname">df</code> är derivatan av <code class="varname">f</code>, och <code
class="varname">ddf</code> är andraderivatan av <code class="varname">f</code>. <code
class="varname">gissning</code> är den ursprungliga gissningen. Funktionen returnerar efter att två på
varandra följande värden är inom <code class="varname">epsilon</code> från varandra, eller efter <code
class="varname">maxn</code> försök, i vilket fall funktionen returnerar sedan <code
class="constant">null</code> vilket indikerar misslyckande.</p><p>Se även <a class="link"
href="ch11s13.html#gel-function-NewtonsMethod"><code class="function">NewtonsMethod</code></a> och <a
class="link" href="ch11s19.html#
gel-function-SymbolicDerivative"><code class="function">SymbolicDerivative</code></a>.</p><p>Exempel för att
hitta kvadratroten av 10: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
-</pre><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre
class="synopsis">NewtonsMethod (f,df,gissning,epsilon,maxn)</pre><p>Hitta nollor med Newtons metod. <code
class="varname">f</code> är funktionen och <code class="varname">df</code> är derivatan av <code
class="varname">f</code>. <code class="varname">gissning</code> är den ursprungliga gissningen. Funktionen
returnerar efter två på varandra följande värden inom <code class="varname">epsilon</code> från varandra,
eller efter <code class="varname">maxn</code> försök, i vilket fall funktionen returnerar <code
class="constant">null</code> vilket indikerar misslyckande.</p><p>Se även <a class="link"
href="ch11s15.html#gel-function-NewtonsMethodPoly"><code class="function">NewtonsMethodPoly</code></a>
och <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Exempel för att hitta kvadratroten av 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
-</pre><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Beräkna rötter för ett polynom (grad 1 till 4) med en av formlerna för sådana polynom. Polynomet
ska anges som en vektor av koefficienter. Det vill säga <strong class="userinput"><code>4*x^3 + 2*x +
1</code></strong> motsvarar vektorn <strong class="userinput"><code>[1,2,0,4]</code></strong>. Returnerar en
kolumnvektor av lösningarna.</p><p>Funktionsanropen <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> och <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-func
tion-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre class="synopsis">QuadraticFormula
(p)</pre><p>Beräkna rötter för ett andragradspolynom med formel. Polynomet ska anges som en vektor av
koefficienter. Det vill säga <strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> motsvarar
vektorn <strong class="userinput"><code>[1,2,3]</code></strong>. Returnerar en kolumnvektor av de två
lösningarna.</p><p>Se <a class="ulink" href="http://planetmath.org/QuadraticFormula";
target="_top">Planetmath</a> eller <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Beräkna rötter för ett fjärdegradspolynom med formel. Polynomet ska anges som en vektor av
koefficienter. Det vill säga <strong class="userinput"><code>5*x^4 + 2*x + 1</code></
strong> motsvarar vektorn <strong class="userinput"><code>[1,2,0,0,5]</code></strong>. Returnerar en
kolumnvektor av de fyra lösningarna.</p><p>Se <a class="ulink" href="http://planetmath.org/QuarticFormula";
target="_top">Planetmath</a>, <a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a> eller <a class="ulink" href="http://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Använd klassisk icke-adaptiv Runge-Kuttametod av fjärde ordningen för att numeriskt
lösa y'=f(x,y) för initialt <code class="varname">x0</code>, <code class="varname">y0</code> som går till
<code class="varname">x1</code> med <code class="varname">n</code> inkrement, returnerar <code
class="varname">y</code> vid <code class="varname">x1</code>.</p><p>System kan l
ösas genom att helt enkelt låta <code class="varname">y</code> vara en (kolumn)vektor överallt. Det vill
säga, <code class="varname">y0</code> kan vara en vektor i vilket fall <code class="varname">f</code> bör ta
ett tal <code class="varname">x</code> och en vektor av samma storlek som det andra argumentet och bör
returnera en vektor av samma storlek.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/Runge-KuttaMethod.html"; target="_top">Mathworld</a> eller <a class="ulink"
href="http://en.wikipedia.org/wiki/Runge-Kutta_methods"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>Använd klassisk icke-adaptiv Runge-Kuttametod av fjärde ordningen för att numeriskt
lösa y'=f(x,y) för initialt <code class="varname">x0</code>, <code class="varname">y0</code> som går till
<code class="varname">x1</c
ode> med <code class="varname">n</code> inkrement, returnerar en 2×(<strong
class="userinput"><code>n+1</code></strong>)-matris med <code class="varname">x</code>- och <code
class="varname">y</code>-värdena. Lämplig för att koppla ihop med <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> eller <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
+</pre><p>Se <a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html";
target="_top">Mathworld</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.10 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre
class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p>Hitta rot för en funktion med bisektionsmetoden.
<code class="varname">a</code> och <code class="varname">b</code> är det ursprungliga gissningsintervallet,
<strong class="userinput"><code>f(a)</code></strong> och <strong class="userinput"><code>f(b)</code></strong>
måste ha olika tecken. <code class="varname">TOL</code> är den önskade toleransen och <code
class="varname">N</code> är gränsen för hur många iterationer att köra, 0 betyder ingen gräns. Funktionen
returnerar en vektor <strong class="userinput"><code>[lyckad,värde
,iteration]</code></strong>, där <code class="varname">lyckad</code> är ett booleskt värde som indikerar om
den lyckats, <code class="varname">värde</code> är det sista beräknade värdet, och <code
class="varname">iteration</code> är antalet utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre
class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p>Hitta rot för en funktion med regula
falsi-metoden. <code class="varname">a</code> och <code class="varname">b</code> är det ursprungliga
gissningsintervallet, <strong class="userinput"><code>f(a)</code></strong> och <strong
class="userinput"><code>f(b)</code></strong> måste ha olika tecken. <code class="varname">TOL</code> är den
önskade toleransen och <code class="varname">N</code> är gränsen för hur många iterationer att köra, 0
betyder ingen gräns. Funktionen returnerar en vektor <strong class="userinput"><code>[ly
ckad,värde,iteration]</code></strong>, där <code class="varname">lyckad</code> är ett booleskt värde som
indikerar om den lyckats, <code class="varname">värde</code> är det sista beräknade värdet, och <code
class="varname">iteration</code> är antalet utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre
class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p>Hitta rot för en funktion med Mullers
metod. <code class="varname">TOL</code> är den önskade toleransen och <code class="varname">N</code> är
gränsen för antal iterationer att köra, 0 betyder ingen gräns. Funktionen returnerar en vektor <strong
class="userinput"><code>[lyckad,värde,iteration]</code></strong>, där <code class="varname">lyckad</code> är
ett booleskt värde som indikerar om den lyckats, <code class="varname">värde</code> är det sista beräknade
värdet, och <code class="varname">iteratio
n</code> är antalet utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre class="synopsis">FindRootSecant
(f,a,b,TOL,N)</pre><p>Hitta rot för en funktion med sekantmetoden. <code class="varname">a</code> och <code
class="varname">b</code> är det ursprungliga gissningsintervallet, <strong
class="userinput"><code>f(a)</code></strong> och <strong class="userinput"><code>f(b)</code></strong> måste
ha olika tecken. <code class="varname">TOL</code> är den önskade toleransen och <code
class="varname">N</code> är gränsen för hur många iterationer att köra, 0 betyder ingen gräns. Funktionen
returnerar en vektor <strong class="userinput"><code>[lyckad,värde,iteration]</code></strong>, där <code
class="varname">lyckad</code> är ett booleskt värde som indikerar om den lyckats, <code
class="varname">värde</code> är det sista beräknade värdet, och <code class="varname">iteration</code> är anta
let utförda iterationer.</p></dd><dt><span class="term"><a
name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre class="synopsis">HalleysMethod
(f,df,ddf,gissning,epsilon,maxn)</pre><p>Hitta nollpunkter med Halleys metod. <code class="varname">f</code>
är funktionen, <code class="varname">df</code> är derivatan av <code class="varname">f</code>, och <code
class="varname">ddf</code> är andraderivatan av <code class="varname">f</code>. <code
class="varname">gissning</code> är den ursprungliga gissningen. Funktionen returnerar efter att två på
varandra följande värden är inom <code class="varname">epsilon</code> från varandra, eller efter <code
class="varname">maxn</code> försök, i vilket fall funktionen returnerar sedan <code
class="constant">null</code> vilket indikerar misslyckande.</p><p>Se även <a class="link"
href="ch11s13.html#gel-function-NewtonsMethod"><code class="function">NewtonsMethod</code></a> och <a
class="link" href="ch11s19.html
#gel-function-SymbolicDerivative"><code class="function">SymbolicDerivative</code></a>.</p><p>Exempel för
att hitta kvadratroten av 10: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
+</pre><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre
class="synopsis">NewtonsMethod (f,df,gissning,epsilon,maxn)</pre><p>Hitta nollor med Newtons metod. <code
class="varname">f</code> är funktionen och <code class="varname">df</code> är derivatan av <code
class="varname">f</code>. <code class="varname">gissning</code> är den ursprungliga gissningen. Funktionen
returnerar efter två på varandra följande värden inom <code class="varname">epsilon</code> från varandra,
eller efter <code class="varname">maxn</code> försök, i vilket fall funktionen returnerar <code
class="constant">null</code> vilket indikerar misslyckande.</p><p>Se även <a class="link"
href="ch11s15.html#gel-function-NewtonsMethodPoly"><code class="function">NewtonsMethodPoly</code></a>
och <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code
class="function">SymbolicDerivative</code></a>.</p><p>Exempel för att hitta kvadratroten av 10: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
+</pre><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
för mer information.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre class="synopsis">PolynomialRoots
(p)</pre><p>Beräkna rötter för ett polynom (grad 1 till 4) med en av formlerna för sådana polynom. Polynomet
ska anges som en vektor av koefficienter. Det vill säga <strong class="userinput"><code>4*x^3 + 2*x +
1</code></strong> motsvarar vektorn <strong class="userinput"><code>[1,2,0,4]</code></strong>. Returnerar en
kolumnvektor av lösningarna.</p><p>Funktionsanropen <a class="link"
href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>, <a class="link"
href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a> och <a class="link"
href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.</p></dd><dt><span class="term"><a
name="gel-fun
ction-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre class="synopsis">QuadraticFormula
(p)</pre><p>Beräkna rötter för ett andragradspolynom med formel. Polynomet ska anges som en vektor av
koefficienter. Det vill säga <strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> motsvarar
vektorn <strong class="userinput"><code>[1,2,3]</code></strong>. Returnerar en kolumnvektor av de två
lösningarna.</p><p>Se <a class="ulink" href="http://planetmath.org/QuadraticFormula";
target="_top">Planetmath</a>, <a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html";
target="_top">Mathworld</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre class="synopsis">QuarticFormula
(p)</pre><p>Beräkna rötter för ett fjärdegradspolynom med formel. Polynomet ska anges s
om en vektor av koefficienter. Det vill säga <strong class="userinput"><code>5*x^4 + 2*x + 1</code></strong>
motsvarar vektorn <strong class="userinput"><code>[1,2,0,0,5]</code></strong>. Returnerar en kolumnvektor av
de fyra lösningarna.</p><p>Se <a class="ulink" href="http://planetmath.org/QuarticFormula";
target="_top">Planetmath</a>, <a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html";
target="_top">Mathworld</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation";
target="_top">Wikipedia</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre class="synopsis">RungeKutta
(f,x0,y0,x1,n)</pre><p>Använd klassisk icke-adaptiv Runge-Kuttametod av fjärde ordningen för att numeriskt
lösa y'=f(x,y) för initialt <code class="varname">x0</code>, <code class="varname">y0</code> som går till
<code class="varname">x1</code> med <code class="varname">n</code> inkre
ment, returnerar <code class="varname">y</code> vid <code class="varname">x1</code>.</p><p>System kan lösas
genom att helt enkelt låta <code class="varname">y</code> vara en (kolumn)vektor överallt. Det vill säga,
<code class="varname">y0</code> kan vara en vektor i vilket fall <code class="varname">f</code> bör ta ett
tal <code class="varname">x</code> och en vektor av samma storlek som det andra argumentet och bör returnera
en vektor av samma storlek.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/Runge-KuttaMethod.html"; target="_top">Mathworld</a> eller <a class="ulink"
href="https://en.wikipedia.org/wiki/Runge-Kutta_methods"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre class="synopsis">RungeKuttaFull
(f,x0,y0,x1,n)</pre><p>Använd klassisk icke-adaptiv Runge-Kuttametod av fjärde ordningen för att numeriskt
lösa y'=f(x,y) för initialt <c
ode class="varname">x0</code>, <code class="varname">y0</code> som går till <code class="varname">x1</code>
med <code class="varname">n</code> inkrement, returnerar en (<strong
class="userinput"><code>n+1</code></strong>)×2-matris med <code class="varname">x</code>- och <code
class="varname">y</code>-värdena. Lämplig för att koppla ihop med <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> eller <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line =
RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponentiell
tillväxt");</code></strong>
</pre><p>System kan lösas genom att helt enkelt låta <code class="varname">y</code> vara en (kolumn)vektor
överallt. Det vill säga, <code class="varname">y0</code> kan vara en vektor i vilket fall <code
class="varname">f</code> bör ta ett tal <code class="varname">x</code> och en vektor av samma storlek som det
andra argumentet och bör returnera en vektor av samma storlek.</p><p>Utdata för ett system är fortfarande en
n×2-matris där den andra posten är en vektor. Om du vill rita linjen, se till att använda radvektorer och
platta sedan till matrisen med <a class="link" href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>
och välj de rätta kolumnerna. Exempel: </p><pre class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotClear();</code></strong>
@@ -22,4 +22,4 @@
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow =
[0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","Första");</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Andra");</code></strong>
-</pre><p>Se <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> eller <a class="ulink" href="http://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.10 och
framåt.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s12.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s14.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Funktioner
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Statistik</td></tr></table></div></body></html>
+</pre><p>Se <a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html";
target="_top">Mathworld</a> eller <a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods";
target="_top">Wikipedia</a> för mer information.</p><p>Version 1.0.10 och
framåt.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s12.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s14.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Funktioner
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Statistik</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s14.html b/help/sv/html/ch11s14.html
index 5ea2118..f38af0b 100644
--- a/help/sv/html/ch11s14.html
+++ b/help/sv/html/ch11s14.html
@@ -1 +1 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Statistik</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s13.html" title="Ekvationslösning"><link rel="next" href="ch11s15.html"
title="Polynom"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Statistik</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s13.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style
="clear: both"><a name="genius-gel-function-list-statistics"></a>Statistik</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alias: <code
class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Beräkna medelvärde för en hel matris.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral av GaussFunction från 0 till <code
class="varname">x</code> (area under normalkurvan).</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="t
erm"><a name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre
class="synopsis">GaussFunction (x,sigma)</pre><p>Gauss normaliserade distributionsfunktion
(normalkurvan).</p><p>Se <a class="ulink" href="http://mathworld.wolfram.com/NormalDistribution.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Median"></a>Median</span></dt><dd><pre class="synopsis">Median (m)</pre><p>Alias: <code
class="function">median</code></p><p>Beräkna median för en hel matris.</p><p>Se <a class="ulink"
href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">stdevp</code></p><p>Beräkna populationsstandardavvikelsen för en hel
matris.</p></dd><dt><span
class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Alias: <code class="function">RowMean</code></p><p>Beräkna medelvärde
för varje rad i en matris.</p><p>Se <a class="ulink" href="http://mathworld.wolfram.com/ArithmeticMean.html";
target="_top">Mathworld</a> för mer information.</p></dd><dt><span class="term"><a
name="gel-function-RowMedian"></a>RowMedian</span></dt><dd><pre class="synopsis">RowMedian
(m)</pre><p>Beräkna median för varje rad i en matris och returnera en kolumnvektor över medianerna.</p><p>Se
<a class="ulink" href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> för
mer information.</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">rowstdevp</code></p><p>Beräkna populatio
nsstandardavvikelserna för rader i en matris och returnera en vertikal vektor.</p></dd><dt><span
class="term"><a name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alias: <code class="function">rowstdev</code></p><p>Beräkna
standardavvikelserna för rader av en matris och returnera en vertikal vektor.</p></dd><dt><span
class="term"><a name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alias: <code class="function">stdev</code></p><p>Beräkna
standardavvikelsen för en hel matris.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s15.html">N�
�sta</a></td></tr><tr><td width="40%" align="left" valign="top">Ekvationslösning </td><td width="20%"
align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%" align="right" valign="top">
Polynom</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Statistik</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s13.html" title="Ekvationslösning"><link rel="next" href="ch11s15.html"
title="Polynom"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Statistik</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s13.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s15.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style
="clear: both"><a name="genius-gel-function-list-statistics"></a>Statistik</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-Average"></a>Average</span></dt><dd><pre class="synopsis">Average (m)</pre><p>Alias: <code
class="function">average</code><code class="function">Mean</code><code
class="function">mean</code></p><p>Beräkna medel (aritmetiskt medelvärde) för en hel matris.</p><p>Se <a
class="ulink" href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-GaussDistribution"></a>GaussDistribution</span></dt><dd><pre
class="synopsis">GaussDistribution (x,sigma)</pre><p>Integral av GaussFunction från 0 till <code
class="varname">x</code> (area under normalkurvan).</p><p>Se <a class="ulink" href="https://en.wikipe
dia.org/wiki/Normal_distribution" target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-GaussFunction"></a>GaussFunction</span></dt><dd><pre class="synopsis">GaussFunction
(x,sigma)</pre><p>Gauss normaliserade distributionsfunktion (normalkurvan).</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Normal_distribution"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/NormalDistribution.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-Median"></a>Median</span></dt><dd><pre
class="synopsis">Median (m)</pre><p>Alias: <code class="function">median</code></p><p>Beräkna median för en
hel matris.</p><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a>
eller <a class="ulink" hr
ef="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-PopulationStandardDeviation"></a>PopulationStandardDeviation</span></dt><dd><pre
class="synopsis">PopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">stdevp</code></p><p>Beräkna populationsstandardavvikelsen för en hel
matris.</p></dd><dt><span class="term"><a name="gel-function-RowAverage"></a>RowAverage</span></dt><dd><pre
class="synopsis">RowAverage (m)</pre><p>Alias: <code class="function">RowMean</code></p><p>Beräkna medel för
varje rad i en matris. Det vill säga beräkna aritmetiskt medelvärde.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Mean"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/ArithmeticMean.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-RowMedian"></a>RowMed
ian</span></dt><dd><pre class="synopsis">RowMedian (m)</pre><p>Beräkna median för varje rad i en matris och
returnera en kolumnvektor över medianerna.</p><p>Se <a class="ulink"
href="https://en.wikipedia.org/wiki/Median"; target="_top">Wikipedia</a> eller <a class="ulink"
href="http://mathworld.wolfram.com/StatisticalMedian.html"; target="_top">Mathworld</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-RowPopulationStandardDeviation"></a>RowPopulationStandardDeviation</span></dt><dd><pre
class="synopsis">RowPopulationStandardDeviation (m)</pre><p>Alias: <code
class="function">rowstdevp</code></p><p>Beräkna populationsstandardavvikelserna för rader i en matris och
returnera en vertikal vektor.</p></dd><dt><span class="term"><a
name="gel-function-RowStandardDeviation"></a>RowStandardDeviation</span></dt><dd><pre
class="synopsis">RowStandardDeviation (m)</pre><p>Alias: <code class="function">rowstdev</code></p><p>Beräkna
standardavvikelserna f�
�r rader av en matris och returnera en vertikal vektor.</p></dd><dt><span class="term"><a
name="gel-function-StandardDeviation"></a>StandardDeviation</span></dt><dd><pre
class="synopsis">StandardDeviation (m)</pre><p>Alias: <code class="function">stdev</code></p><p>Beräkna
standardavvikelsen för en hel matris.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s13.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s15.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Ekvationslösning
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Polynom</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s15.html b/help/sv/html/ch11s15.html
index ca0c33d..e50ed66 100644
--- a/help/sv/html/ch11s15.html
+++ b/help/sv/html/ch11s15.html
@@ -1,2 +1,2 @@
<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Polynom</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s14.html" title="Statistik"><link rel="next" href="ch11s16.html" title="Mängdlära"></head><body
bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table
width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Polynom</th></tr><tr><td
width="20%" align="left"><a accesskey="p" href="ch11s14.html">Föregående</a> </td><th width="60%"
align="center">Kapitel 11. Lista över GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s16.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="title" style="clear:
both"><a name="genius-gel-function-list-polynomials"></a>Polynom</h2></div></div></div><div
class="variablelist"><dl class="variablelist"><dt><span class="term"><a
name="gel-function-AddPoly"></a>AddPoly</span></dt><dd><pre class="synopsis">AddPoly (p1,p2)</pre><p>Addera
två polynom (vektorer).</p></dd><dt><span class="term"><a
name="gel-function-DividePoly"></a>DividePoly</span></dt><dd><pre class="synopsis">DividePoly
(p,q,&r)</pre><p>Dividera två polynom (som vektorer) med lång division. Returnerar kvoten av de två
polynomen. Det valfria argumentet <code class="varname">r</code> används för att returnera resten. Rester
kommer ha lägre grad än <code class="varname">q</code>.</p><p>Se <a class="ulink"
href="http://planetmath.org/PolynomialLongDivision"; target="_top">Planetmath</a> för mer
information.</p></dd><dt><span class="term"><a name="gel-function-IsPoly"></a>IsPoly</span></dt><dd><pre
class="synopsis">IsPoly (p)</pre><p>Kontrollera om en vektor är anv�
�ndbar som ett polynom.</p></dd><dt><span class="term"><a
name="gel-function-MultiplyPoly"></a>MultiplyPoly</span></dt><dd><pre class="synopsis">MultiplyPoly
(p1,p2)</pre><p>Multiplicera två polynom (som vektorer).</p></dd><dt><span class="term"><a
name="gel-function-NewtonsMethodPoly"></a>NewtonsMethodPoly</span></dt><dd><pre
class="synopsis">NewtonsMethodPoly (poly,gissning,epsilon,maxn)</pre><p>Hitta en rot av ett polynom med
Newtons metod. <code class="varname">poly</code> är polynomet som en vektor och <code
class="varname">gissning</code> är den ursprungliga gissningen. Funktionen returnerar efter två på varandra
följande värden inom <code class="varname">epsilon</code> från varandra, eller efter <code
class="varname">maxn</code> försök, i vilket fall funktionen returnerar <code class="constant">null</code>
vilket indikerar misslyckande.</p><p>Se även <a class="link"
href="ch11s13.html#gel-function-NewtonsMethod"><code class="function">NewtonsMethod</code></
a>.</p><p>Exempel för att hitta kvadratroten av 10: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>NewtonsMethodPoly([-10,0,1],3,10^-10,100)</code></strong>
-</pre><p>Se <a class="ulink" href="http://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Ta andraderivata av polynom (som vektor).</p></dd><dt><span
class="term"><a name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre
class="synopsis">PolyDerivative (p)</pre><p>Ta derivata av polynom (som vektor).</p></dd><dt><span
class="term"><a name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre
class="synopsis">PolyToFunction (p)</pre><p>Skapa funktion av ett polynom (som en vektor).</p></dd><dt><span
class="term"><a name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre
class="synopsis">PolyToString (p,var...)</pre><p>Skapa sträng av ett polynom (som en
vektor).</p></dd><dt><span class="term"><a name="gel-function-SubtractPoly"></
a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly (p1,p2)</pre><p>Subtrahera två polynom (som
vektorer).</p></dd><dt><span class="term"><a name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre
class="synopsis">TrimPoly (p)</pre><p>Ta bort nollor från ett polynom (som
vektor).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s14.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s16.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Statistik
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Mängdlära</td></tr></table></div></body></html>
+</pre><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method"; target="_top">Wikipedia</a>
för mer information.</p></dd><dt><span class="term"><a
name="gel-function-Poly2ndDerivative"></a>Poly2ndDerivative</span></dt><dd><pre
class="synopsis">Poly2ndDerivative (p)</pre><p>Ta andraderivata av polynom (som vektor).</p></dd><dt><span
class="term"><a name="gel-function-PolyDerivative"></a>PolyDerivative</span></dt><dd><pre
class="synopsis">PolyDerivative (p)</pre><p>Ta derivata av polynom (som vektor).</p></dd><dt><span
class="term"><a name="gel-function-PolyToFunction"></a>PolyToFunction</span></dt><dd><pre
class="synopsis">PolyToFunction (p)</pre><p>Skapa funktion av ett polynom (som en vektor).</p></dd><dt><span
class="term"><a name="gel-function-PolyToString"></a>PolyToString</span></dt><dd><pre
class="synopsis">PolyToString (p,var...)</pre><p>Skapa sträng av ett polynom (som en
vektor).</p></dd><dt><span class="term"><a name="gel-function-SubtractPoly"><
/a>SubtractPoly</span></dt><dd><pre class="synopsis">SubtractPoly (p1,p2)</pre><p>Subtrahera två polynom
(som vektorer).</p></dd><dt><span class="term"><a
name="gel-function-TrimPoly"></a>TrimPoly</span></dt><dd><pre class="synopsis">TrimPoly (p)</pre><p>Ta bort
nollor från ett polynom (som vektor).</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%"
summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s14.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s16.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Statistik </td><td
width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%" align="right"
valign="top"> Mängdlära</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s18.html b/help/sv/html/ch11s18.html
index 94cd702..65e19e8 100644
--- a/help/sv/html/ch11s18.html
+++ b/help/sv/html/ch11s18.html
@@ -1 +1,13 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Diverse</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s17.html" title="Kommutativ algebra"><link rel="next" href="ch11s19.html" title="Symboliska
operationer"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Diverse</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="
title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Diverse</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vek)</pre><p>Konvertera en vektor med ASCII-värden till en sträng.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vek,alfabet)</pre><p>Konvertera en vektor med 0-baserade alfabetvärden
(positioner i alfabetsträngen) till en sträng.</p></dd><dt><span class="term"><a
name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre class="synopsis">StringToASCII
(str)</pre><p>Konvertera en sträng till en vektor med ASCII-värden.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alf
abet)</pre><p>Konvertera en sträng till en vektor med 0-baserade alfabetvärden (positioner i
alfabetsträngen), -1 för okända bokstäver.</p></dd></dl></div></div><div class="navfooter"><hr><table
width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p"
href="ch11s17.html">Föregående</a> </td><td width="20%" align="center"><a accesskey="u"
href="ch11.html">Upp</a></td><td width="40%" align="right"> <a accesskey="n"
href="ch11s19.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Kommutativ algebra
</td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Symboliska operationer</td></tr></table></div></body></html>
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Diverse</title><meta
name="generator" content="DocBook XSL Stylesheets V1.79.1"><link rel="home" href="index.html" title="Handbok
för Genius"><link rel="up" href="ch11.html" title="Kapitel 11. Lista över GEL-funktioner"><link rel="prev"
href="ch11s17.html" title="Kommutativ algebra"><link rel="next" href="ch11s19.html" title="Symboliska
operationer"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div
class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3"
align="center">Diverse</th></tr><tr><td width="20%" align="left"><a accesskey="p"
href="ch11s17.html">Föregående</a> </td><th width="60%" align="center">Kapitel 11. Lista över
GEL-funktioner</th><td width="20%" align="right"> <a accesskey="n"
href="ch11s19.html">Nästa</a></td></tr></table><hr></div><div class="sect1"><div
class="titlepage"><div><div><h2 class="
title" style="clear: both"><a
name="genius-gel-function-list-miscellaneous"></a>Diverse</h2></div></div></div><div class="variablelist"><dl
class="variablelist"><dt><span class="term"><a
name="gel-function-ASCIIToString"></a>ASCIIToString</span></dt><dd><pre class="synopsis">ASCIIToString
(vek)</pre><p>Konvertera en vektor av ASCII-värden till en sträng. Se även <a class="link"
href="ch11s18.html#gel-function-StringToASCII"><code
class="function">StringToASCII</code></a>.</p><p>Exempel: </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>ASCIIToString([97,98,99])</code></strong>
+= "abc"
+</pre><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-AlphabetToString"></a>AlphabetToString</span></dt><dd><pre
class="synopsis">AlphabetToString (vek,alfabet)</pre><p>Konvertera en vektor med alfabetsvärdena 0 och uppåt
(positioner i alfabetssträngen) till en sträng. En <code class="constant">null</code>-vektor resulterar i en
tom sträng. Se även <a class="link" href="ch11s18.html#gel-function-StringToAlphabet"><code
class="function">StringToAlphabet</code></a>.</p><p>Exempel: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString([1,2,3,0,0],"abcd")</code></strong>
+= "bcdaa"
+<code class="prompt">genius></code> <strong
class="userinput"><code>AlphabetToString(null,"abcd")</code></strong>
+= ""
+</pre></dd><dt><span class="term"><a name="gel-function-StringToASCII"></a>StringToASCII</span></dt><dd><pre
class="synopsis">StringToASCII (str)</pre><p>Konvertera en sträng till en (rad)vektor av ASCII-värden. Se
även <a class="link" href="ch11s18.html#gel-function-ASCIIToString"><code
class="function">ASCIIToString</code></a>.</p><p>Exempel: </p><pre class="screen"><code
class="prompt">genius></code> <strong class="userinput"><code>StringToASCII("abc")</code></strong>
+= [97, 98, 99]
+</pre><p>Se <a class="ulink" href="https://en.wikipedia.org/wiki/ASCII"; target="_top">Wikipedia</a> för mer
information.</p></dd><dt><span class="term"><a
name="gel-function-StringToAlphabet"></a>StringToAlphabet</span></dt><dd><pre
class="synopsis">StringToAlphabet (str,alfabet)</pre><p>Konvertera en sträng till en (rad)vektor med
alfabetsvärden 0 och uppåt (positioner i alfabetssträngen), -1 för okända bokstäver. En tom sträng resulterar
i <code class="constant">null</code>. Se även <a class="link"
href="ch11s18.html#gel-function-AlphabetToString"><code
class="function">AlphabetToString</code></a>.</p><p>Exempel: </p><pre class="screen"><code
class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("cca","abcd")</code></strong>
+= [2, 2, 0]
+<code class="prompt">genius></code> <strong
class="userinput"><code>StringToAlphabet("ccag","abcd")</code></strong>
+= [2, 2, 0, -1]
+</pre></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s17.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch11s19.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Kommutativ
algebra </td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td width="40%"
align="right" valign="top"> Symboliska operationer</td></tr></table></div></body></html>
diff --git a/help/sv/html/ch11s20.html b/help/sv/html/ch11s20.html
index 8e27854..f20b16e 100644
--- a/help/sv/html/ch11s20.html
+++ b/help/sv/html/ch11s20.html
@@ -2,15 +2,15 @@
<code class="prompt">genius></code> <strong
class="userinput"><code>ExportPlot("/katalog/fil","eps")</code></strong>
</pre><p>Version 1.0.16 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-LinePlot"></a>LinePlot</span></dt><dd><pre class="synopsis">LinePlot
(funk1,funk2,funk3,...)</pre><pre class="synopsis">LinePlot (funk1,funk2,funk3,x1,x2)</pre><pre
class="synopsis">LinePlot (funk1,funk2,funk3,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlot
(funk1,funk2,funk3,[x1,x2])</pre><pre class="synopsis">LinePlot
(funk1,funk2,funk3,[x1,x2,y1,y2])</pre><p>Rita en funktion (eller flera) med en linjegraf. De första (upp
till 10) argumenten är funktioner, sedan kan du valfritt ange gränserna för graffönstret som <code
class="varname">x1</code>, <code class="varname">x2</code>, <code class="varname">y1</code>, <code
class="varname">y2</code>. Om gränser inte anges kommer de aktuellt inställda gränserna att användas (Se <a
class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>) Om y-gränserna inte anges beräknas funktio
nerna och sedan används max- och minvärdena.</p><p>Parametern <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
styr om förklaringen ritas ut.</p><p>Exempel: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>LinePlot(sin,cos)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlot(`(x)=x^2,-1,1,0,1)</code></strong>
-</pre></dd><dt><span class="term"><a name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre
class="synopsis">LinePlotClear ()</pre><p>Visa linjegrafsfönstret och rensa bort funktioner och alla andra
linjer som ritades.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (funk,...)</pre><pre class="synopsis">LinePlotCParametric
(funk,t1,t2,tinc)</pre><pre class="synopsis">LinePlotCParametric (funk,t1,t2,tinc,x1,x2,y1,y2)</pre><p>Rita
en parametrisk komplexvärd funktion med en linjegraf. Först kommer funktionen som returnerar <code
class="computeroutput">x+iy</code> sedan valfritt <code class="varname">t</code>-gränserna som <strong
class="userinput"><code>t1,t2,tinc</code></strong>, sedan valfritt gränserna som <strong
class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Om gränser inte anges kommer de aktuellt inställda
gränserna att användas
(Se <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>). Om istället strängen "fit" anges för x- och y-gränserna kommer
gränserna vara den största utsträckningen för grafen</p><p>Parametern <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
styr om förklaringen ritas ut.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>Rita en linje från <code class="varname">x1</code>,<code class="varname">y1</code> till <code
class="varname">x2</code>,<code class="varname">y2</code>. <code class="varname">x1</code>,<code
class="varname">y1</code>, <code class="varname">x2</code>,<code class="varname">y2</code> kan ersättas med
en <code class="varname">n</code>×2-matris för
ett längre polygontåg. Alternativt kan vektorn <code class="varname">v</code> vara en kolumnvektor med
komplexa tal, det vill säga en <code class="varname">n</code>×1-matris och varje komplext tal anses då vara
en punkt i planet.</p><p>Extra parametrar kan läggas till för att ange linjefärg, tjocklek, pilar,
graffönstret eller förklaring. Du kan göra detta genom att lägga till en argumentsträng <strong
class="userinput"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>, <strong
class="userinput"><code>"arrow"</code></strong> eller <strong
class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen, tjockleken, fönstret som en
4-vektor, piltyp eller förklaringen. (Pil och fönster är från version 1.0.6 och framåt.)</p><p>Om linjen ska
behandlas som en fylld polygon som är fylld med den angivna färgen kan du ange argumentet <strong class="u
serinput"><code>"filled"</code></strong>. Sedan version 1.0.22 och framåt.</p><p>Färgen ska vara antingen en
sträng som indikerar det vanliga engelska ordet för färgen som GTK kommer känna igen, som <strong
class="userinput"><code>"red"</code></strong>, <strong class="userinput"><code>"blue"</code></strong>,
<strong class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format
som <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan sedan version 1.0.18 färgen också anges som en reell
vektor som anger de röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1, t.ex. <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Fönstret ska som vanligt anges som <st
rong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem percents gränser kring linjen.</p><p>Pilspecifikation ska vara <strong
class="userinput"><code>"origin"</code></strong>, <strong class="userinput"><code>"end"</code></strong>,
<strong class="userinput"><code>"both"</code></strong> eller <strong
class="userinput"><code>"none"</code></strong>.</p><p>Slutligen ska förklaring vara en sträng som kan
användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
+</pre></dd><dt><span class="term"><a name="gel-function-LinePlotClear"></a>LinePlotClear</span></dt><dd><pre
class="synopsis">LinePlotClear ()</pre><p>Visa linjegrafsfönstret och rensa bort funktioner och alla andra
linjer som ritades.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotCParametric"></a>LinePlotCParametric</span></dt><dd><pre
class="synopsis">LinePlotCParametric (funk,...)</pre><pre class="synopsis">LinePlotCParametric
(funk,t1,t2,tinc)</pre><pre class="synopsis">LinePlotCParametric (funk,t1,t2,tinc,x1,x2,y1,y2)</pre><p>Rita
en parametrisk komplexvärd funktion med en linjegraf. Först kommer funktionen som returnerar <code
class="computeroutput">x+iy</code> sedan valfritt <code class="varname">t</code>-gränserna som <strong
class="userinput"><code>t1,t2,tinc</code></strong>, sedan valfritt gränserna som <strong
class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Om gränser inte anges kommer de aktuellt inställda
gränserna att användas
(Se <a class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>). Om istället strängen "fit" anges för x- och y-gränserna kommer
gränserna vara den största utsträckningen för grafen</p><p>Parametern <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
styr om förklaringen ritas ut.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawLine"></a>LinePlotDrawLine</span></dt><dd><pre
class="synopsis">LinePlotDrawLine (x1,y1,x2,y2,...)</pre><pre class="synopsis">LinePlotDrawLine
(v,...)</pre><p>Rita en linje från <code class="varname">x1</code>,<code class="varname">y1</code> till <code
class="varname">x2</code>,<code class="varname">y2</code>. <code class="varname">x1</code>,<code
class="varname">y1</code>, <code class="varname">x2</code>,<code class="varname">y2</code> kan ersättas med
en <code class="varname">n</code>×2-matris för
ett längre polygontåg. Alternativt kan vektorn <code class="varname">v</code> vara en kolumnvektor med
komplexa tal, det vill säga en <code class="varname">n</code>×1-matris och varje komplext tal anses då vara
en punkt i planet.</p><p>Extra parametrar kan läggas till för att ange linjefärg, tjocklek, pilar,
graffönstret eller förklaring. Du kan göra detta genom att lägga till en argumentsträng <strong
class="userinput"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong>, <strong
class="userinput"><code>"arrow"</code></strong> eller <strong
class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen, tjockleken, fönstret som en
4-vektor, piltyp eller förklaringen. (Pil och fönster är från version 1.0.6 och framåt.)</p><p>Om linjen ska
behandlas som en fylld polygon som är fylld med den angivna färgen kan du ange argumentet <strong class="u
serinput"><code>"filled"</code></strong>. Sedan version 1.0.22 och framåt.</p><p>Färgen ska vara antingen en
sträng som indikerar det vanliga engelska ordet för färgen som GTK kommer känna igen, som <strong
class="userinput"><code>"red"</code></strong>, <strong class="userinput"><code>"blue"</code></strong>,
<strong class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format
som <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan sedan version 1.0.18 färgen också anges som en reell
vektor som anger de röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1, t.ex. <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Fönstret ska som vanligt anges som <st
rong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem procents gränser kring linjen.</p><p>Pilspecifikation ska vara <strong
class="userinput"><code>"origin"</code></strong>, <strong class="userinput"><code>"end"</code></strong>,
<strong class="userinput"><code>"both"</code></strong> eller <strong
class="userinput"><code>"none"</code></strong>.</p><p>Slutligen ska förklaring vara en sträng som kan
användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,-1;-1,-1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;1,1],"arrow","end")</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","Lösningen")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","Lösningen")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>for r=0.0 to 1.0 by 0.1 do
LinePlotDrawLine([0,0;1,r],"color",[r,(1-r),0.5],"window",[0,1,0,1])</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")</code></strong>
-</pre><p>Till skillnad från många andra funktioner som inte bryr sig om de tar en kolumn- eller radvektor så
måste på grund av möjliga tvetydigheter punkter som anges som en vektor av komplexa tal alltid anges som en
kolumnvektor.</p><p>Att ange <code class="varname">v</code> som en kolumnvektor med komplexa tal är
implementerat från version 1.0.22 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints
(v,...)</pre><p>Rita en punkt vid <code class="varname">x</code>,<code class="varname">y</code>. Indata kan
vara en <code class="varname">n</code>×2-matris för <code class="varname">n</code> olika punkter. Denna
funktion har i stort sett samma indata som <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>. Alternativt kan vektorn <code
class="varname">v</code> vara en k
olumnvektor med komplexa tal, det vill säga en <code class="varname">n</code>×1-matris och varje komplext
tal anses då vara en punkt i planet.</p><p>Extra parametrar kan läggas till för att ange färg, tjocklek,
graffönstret eller förklaring. Du kan göra detta genom att lägga till en argumentsträng <strong
class="userinput"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong> eller <strong
class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen, tjockleken, fönstret som en
4-vektor eller förklaringen.</p><p>Färgen ska vara antingen en sträng som indikerar det vanliga engelska
ordet för färgen som GTK kommer känna igen, som <strong class="userinput"><code>"red"</code></strong>,
<strong class="userinput"><code>"blue"</code></strong>, <strong
class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format s
om <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan färgen också anges som en reell vektor som anger de
röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1.</p><p>Fönstret ska som vanligt anges
som <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem percents gränser kring linjen.</p><p>Slutligen ska förklaring vara en sträng som
kan användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>Lin
ePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
+</pre><p>Till skillnad från många andra funktioner som inte bryr sig om de tar en kolumn- eller radvektor så
måste på grund av möjliga tvetydigheter punkter som anges som en vektor av komplexa tal alltid anges som en
kolumnvektor.</p><p>Att ange <code class="varname">v</code> som en kolumnvektor med komplexa tal är
implementerat från version 1.0.22 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotDrawPoints"></a>LinePlotDrawPoints</span></dt><dd><pre
class="synopsis">LinePlotDrawPoints (x,y,...)</pre><pre class="synopsis">LinePlotDrawPoints
(v,...)</pre><p>Rita en punkt vid <code class="varname">x</code>,<code class="varname">y</code>. Indata kan
vara en <code class="varname">n</code>×2-matris för <code class="varname">n</code> olika punkter. Denna
funktion har i stort sett samma indata som <a class="link"
href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a>. Alternativt kan vektorn <code
class="varname">v</code> vara en k
olumnvektor med komplexa tal, det vill säga en <code class="varname">n</code>×1-matris och varje komplext
tal anses då vara en punkt i planet.</p><p>Extra parametrar kan läggas till för att ange färg, tjocklek,
graffönstret eller förklaring. Du kan göra detta genom att lägga till en argumentsträng <strong
class="userinput"><code>"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>,
<strong class="userinput"><code>"window"</code></strong> eller <strong
class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen, tjockleken, fönstret som en
4-vektor eller förklaringen.</p><p>Färgen ska vara antingen en sträng som indikerar det vanliga engelska
ordet för färgen som GTK kommer känna igen, som <strong class="userinput"><code>"red"</code></strong>,
<strong class="userinput"><code>"blue"</code></strong>, <strong
class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format s
om <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan färgen också anges som en reell vektor som anger de
röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1.</p><p>Fönstret ska som vanligt anges
som <strong class="userinput"><code>[x1,x2,y1,y2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem procents gränser kring linjen.</p><p>Slutligen ska förklaring vara en sträng som
kan användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>Lin
ePlotDrawPoints(0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([0,0;1,-1;-1,-1])</code></strong>
-<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,3,100),"color","blue","legend","Lösningen")</code></strong>
+<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color","blue","legend","Lösningen")</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7)),"thickness",3,"legend","7:e
enhetsrötterna")</code></strong>
</pre><p>Till skillnad från många andra funktioner som inte bryr sig om de tar en kolumn- eller radvektor så
måste på grund av möjliga tvetydigheter punkter som anges som en vektor av komplexa tal alltid anges som en
kolumnvektor. Notera därför i sista exemplet transponatet av vektorn <strong
class="userinput"><code>0:6</code></strong> för att göra den till en kolumvektor.</p><p>Tillgängligt från
version 1.0.18 och framåt. Att ange <code class="varname">v</code> som en kolumnvektor med komplexa tal är
implementerat från version 1.0.22 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotMouseLocation"></a>LinePlotMouseLocation</span></dt><dd><pre
class="synopsis">LinePlotMouseLocation ()</pre><p>Returnerar en radvektor för en punkt på linjegrafen som
motsvarar den aktuella positionen för musen. Om linjegrafen inte är synlig skrivs ett fel ut och <code
class="constant">null</code> returneras. I detta fall bör du köra <a class="link" h
ref="ch11s20.html#gel-function-LinePlot"><code class="function">LinePlot</code></a> eller <a class="link"
href="ch11s20.html#gel-function-LinePlotClear"><code class="function">LinePlotClear</code></a> för att ställa
graffönstret i linjegrafsläget. Se även <a class="link"
href="ch11s20.html#gel-function-LinePlotWaitForClick"><code
class="function">LinePlotWaitForClick</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-LinePlotParametric"></a>LinePlotParametric</span></dt><dd><pre
class="synopsis">LinePlotParametric (xfunk,yfunk,...)</pre><pre class="synopsis">LinePlotParametric
(xfunk,yfunk,t1,t2,tinc)</pre><pre class="synopsis">LinePlotParametric
(xfunk,yfunk,t1,t2,tinc,x1,x2,y1,y2)</pre><pre class="synopsis">LinePlotParametric
(xfunk,yfunk,t1,t2,tinc,[x1,x2,y1,y2])</pre><pre class="synopsis">LinePlotParametric
(xfunk,yfunk,t1,t2,tinc,"fit")</pre><p>Rita en parametrisk funktion med en linjegraf. Först kommer
funktionerna för <code class="varname">x</code
och <code class="varname">y</code>, sedan valfritt <code class="varname">t</code>-gränserna som <strong
class="userinput"><code>t1,t2,tinc</code></strong>, sedan valfritt gränserna som <strong
class="userinput"><code>x1,x2,y1,y2</code></strong>.</p><p>Om x- och y-gränser inte anges kommer de
aktuellt inställda gränserna att användas (Se <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">LinePlotWindow</code></a>). Om
istället strängen "fit" anges för x- och y-gränserna kommer gränserna vara den största utsträckningen för
grafen</p><p>Parametern <a class="link" href="ch11s03.html#gel-function-LinePlotDrawLegends"><code
class="function">LinePlotDrawLegends</code></a> styr om förklaringen ritas ut.</p></dd><dt><span
class="term"><a name="gel-function-LinePlotWaitForClick"></a>LinePlotWaitForClick</span></dt><dd><pre
class="synopsis">LinePlotWaitForClick ()</pre><p>Om i linjegrafsläge så inväntas ett klick på linjegrafsfö
nstret och platsen för klicket returneras som en radvektor. Om fönstret är stängt returnerar funktionen
omedelbart <code class="constant">null</code>. Om fönstret inte är i linjegrafsläge ställs det i det läget
och visas om det inte visats. Se även <a class="link"
href="ch11s20.html#gel-function-LinePlotMouseLocation"><code
class="function">LinePlotMouseLocation</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasFreeze"></a>PlotCanvasFreeze</span></dt><dd><pre
class="synopsis">PlotCanvasFreeze ()</pre><p>Frys tillfälligt ritande av grafens rityta. Användbart om du
behöver rita ett gäng element och vill fördröja ritande av allt för att undvika flimmer i en animering. Efter
att allt har ritats bör du anropa <a class="link" href="ch11s20.html#gel-function-PlotCanvasThaw"><code
class="function">PlotCanvasThaw</code></a>.</p><p>Ritytan töas alltid upp efter att en exekvering avslutas.
så den kommer aldrig att förbli frusen. Till exempe
l töas ritytan automatiskt ögonblicket då en ny kommandorad visas. Observera också att anrop för att frysa
och töa upp kan nästas på ett säkert sätt.</p><p>Version 1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-PlotCanvasThaw"></a>PlotCanvasThaw</span></dt><dd><pre class="synopsis">PlotCanvasThaw
()</pre><p>Töa upp ritytan för graf som frystes av <a class="link"
href="ch11s20.html#gel-function-PlotCanvasFreeze"><code class="function">PlotCanvasFreeze</code></a> och rita
om ritytan omedelbart. Ritytan töas också alltid upp efter att ett program slutat exekvera.</p><p>Version
1.0.18 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-PlotWindowPresent"></a>PlotWindowPresent</span></dt><dd><pre
class="synopsis">PlotWindowPresent ()</pre><p>Visa och höj graffönstret, skapa det om nödvändigt. Normalt
skapas fönstret då en av graffunktionerna anropas, men det höjs inte alltid om det råkar vara under andra
fönster. Denn
a funktion är därför bra att anropa i skript där graffönstret kan ha skapats tidigare, och nu är dolt bakom
konsolen eller andra fönster.</p><p>Version 1.0.19 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldClearSolutions"></a>SlopefieldClearSolutions</span></dt><dd><pre
class="synopsis">SlopefieldClearSolutions ()</pre><p>Rensar bort lösningarna som ritats av funktionen <a
class="link" href="ch11s20.html#gel-function-SlopefieldDrawSolution"><code
class="function">SlopefieldDrawSolution</code></a>.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldDrawSolution"></a>SlopefieldDrawSolution</span></dt><dd><pre
class="synopsis">SlopefieldDrawSolution (x, y, dx)</pre><p>Då en riktningsfältsgraf är aktiv, rita en lösning
med de angivna startvillkoren. Den vanliga Runge-Kutta-metoden används med ökning <code
class="varname">dx</code>. Lösningarna stannar i grafen tills en annan graf visas eller tills du anropar <a
class="link"
href="ch11s20.html#gel-function-SlopefieldClearSolutions"><code
class="function">SlopefieldClearSolutions</code></a>. Du kan också använda det grafiska gränssnittet för att
rita lösningar och ange startvillkor med musen.</p></dd><dt><span class="term"><a
name="gel-function-SlopefieldPlot"></a>SlopefieldPlot</span></dt><dd><pre class="synopsis">SlopefieldPlot
(funk)</pre><pre class="synopsis">SlopefieldPlot (funk,x1,x2,y1,y2)</pre><p>Rita ett riktningsfält.
Funktionen <code class="varname">funk</code> ska ta två reella tal <code class="varname">x</code> och <code
class="varname">y</code> eller ett ensamt komplext tal. Valfritt kan du ange gränserna för graffönstret som
<code class="varname">x1</code>, <code class="varname">x2</code>, <code class="varname">y1</code>, <code
class="varname">y2</code>. Om gränser inte anges kommer de aktuellt inställda gränserna att användas (Se <a
class="link" href="ch11s03.html#gel-function-LinePlotWindow"><code class="function">
LinePlotWindow</code></a>).</p><p>Parametern <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
styr om förklaringen ritas ut.</p><p>Exempel: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)</code></strong>
@@ -27,9 +27,9 @@
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(data,[-1,1,-1,1],"Mina data")</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>d:=null; for i=1 to 20 do for j=1 to
10 do d@(i,j) = (0.1*i-1)^2-(0.1*j)^2;</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDataGrid(d,[-1,1,0,1],"halv sadel")</code></strong>
-</pre><p>Version 1.0.16 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>Rita en linje från <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> till <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code>. <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code>, <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code> kan ersättas med en <code class="varname">n</code>×3-matris för ett längre
polygontåg.</p><p>Extra parametrar kan läggas till för att ange linjefärg, tjocklek, pilar, graffönster eller
förklaring. Du kan göra detta genom att lägga till en argumentsträng <strong class="userinput"><code>
"color"</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>, <strong
class="userinput"><code>"window"</code></strong> eller <strong
class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen, tjockleken, fönstret som en
6-vektor eller förklaringen.</p><p>Färgen ska vara antingen en sträng som indikerar det vanliga engelska
ordet för färgen som GTK kommer känna igen, som <strong class="userinput"><code>"red"</code></strong>,
<strong class="userinput"><code>"blue"</code></strong>, <strong
class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format som
<strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan sedan version 1.0.18 färgen också anges som
en reell vektor som anger de röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1, t.ex.
<strong class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Fönstret ska som vanligt anges som
<strong class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, eller kan alternativt anges som en
sträng <strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in
precis och y-intervallet med fem percents gränser kring linjen.</p><p>Slutligen ska förklaring vara en sträng
som kan användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
+</pre><p>Version 1.0.16 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawLine"></a>SurfacePlotDrawLine</span></dt><dd><pre
class="synopsis">SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)</pre><pre class="synopsis">SurfacePlotDrawLine
(v,...)</pre><p>Rita en linje från <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code> till <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code>. <code class="varname">x1</code>,<code class="varname">y1</code>,<code
class="varname">z1</code>, <code class="varname">x2</code>,<code class="varname">y2</code>,<code
class="varname">z2</code> kan ersättas med en <code class="varname">n</code>×3-matris för ett längre
polygontåg.</p><p>Extra parametrar kan läggas till för att ange linjefärg, tjocklek, graffönster eller
förklaring. Du kan göra detta genom att lägga till en argumentsträng <strong class="userinput"><code>"color"
</code></strong>, <strong class="userinput"><code>"thickness"</code></strong>, <strong
class="userinput"><code>"window"</code></strong> eller <strong
class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen, tjockleken, fönstret som en
6-vektor eller förklaringen.</p><p>Färgen ska vara antingen en sträng som indikerar det vanliga engelska
ordet för färgen som GTK kommer känna igen, som <strong class="userinput"><code>"red"</code></strong>,
<strong class="userinput"><code>"blue"</code></strong>, <strong
class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format som
<strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan sedan version 1.0.18 färgen också anges som en ree
ll vektor som anger de röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1, t.ex. <strong
class="userinput"><code>[1.0,0.5,0.1]</code></strong>.</p><p>Fönstret ska som vanligt anges som <strong
class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem procents gränser kring linjen.</p><p>Slutligen ska förklaring vara en sträng som
kan användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawLine([0,0,0;1,-1,2;-1,-1,-3])</code></strong>
-</pre><p>Tillgängligt i version 1.0.19 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawPoints"></a>SurfacePlotDrawPoints</span></dt><dd><pre
class="synopsis">SurfacePlotDrawPoints (x,y,z,...)</pre><pre class="synopsis">SurfacePlotDrawPoints
(v,...)</pre><p>Rita en punkt vid <code class="varname">x</code>,<code class="varname">y</code>,<code
class="varname">z</code>. Indata kan vara en <code class="varname">n</code>×3-matris för <code
class="varname">n</code> olika punkter. Denna funktion har i huvudsak samma indata som <a class="link"
href="ch11s20.html#gel-function-SurfacePlotDrawLine">SurfacePlotDrawLine</a>.</p><p>Extra parametrar kan
läggas till för att ange linjefärg, tjocklek, graffönster eller förklaring. Du kan göra detta genom att lägga
till en argumentsträng <strong class="userinput"><code>"color"</code></strong>, <strong
class="userinput"><code>"thickness"</code></strong>, <strong class="userinput"><code>"window"</code><
/strong> eller <strong class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen,
tjockleken, fönstret som en 6-vektor eller förklaringen.</p><p>Färgen ska vara antingen en sträng som
indikerar det vanliga engelska ordet för färgen som GTK kommer känna igen, som <strong
class="userinput"><code>"red"</code></strong>, <strong class="userinput"><code>"blue"</code></strong>,
<strong class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format
som <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan färgen också anges som en reell vektor som anger de
röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1.</p><p>Fönstret ska som vanligt anges
som <strong
class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem percents gränser kring linjen.</p><p>Slutligen ska förklaring vara en sträng som
kan användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)</code></strong>
+</pre><p>Tillgängligt i version 1.0.19 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-SurfacePlotDrawPoints"></a>SurfacePlotDrawPoints</span></dt><dd><pre
class="synopsis">SurfacePlotDrawPoints (x,y,z,...)</pre><pre class="synopsis">SurfacePlotDrawPoints
(v,...)</pre><p>Rita en punkt vid <code class="varname">x</code>,<code class="varname">y</code>,<code
class="varname">z</code>. Indata kan vara en <code class="varname">n</code>×3-matris för <code
class="varname">n</code> olika punkter. Denna funktion har i huvudsak samma indata som <a class="link"
href="ch11s20.html#gel-function-SurfacePlotDrawLine">SurfacePlotDrawLine</a>.</p><p>Extra parametrar kan
läggas till för att ange linjefärg, tjocklek, graffönster eller förklaring. Du kan göra detta genom att lägga
till en argumentsträng <strong class="userinput"><code>"color"</code></strong>, <strong
class="userinput"><code>"thickness"</code></strong>, <strong class="userinput"><code>"window"</code><
/strong> eller <strong class="userinput"><code>"legend"</code></strong>, och efter detta ange färgen,
tjockleken, fönstret som en 6-vektor eller förklaringen.</p><p>Färgen ska vara antingen en sträng som
indikerar det vanliga engelska ordet för färgen som GTK kommer känna igen, som <strong
class="userinput"><code>"red"</code></strong>, <strong class="userinput"><code>"blue"</code></strong>,
<strong class="userinput"><code>"yellow"</code></strong>, o.s.v... Alternativt kan färgen anges i RGB-format
som <strong class="userinput"><code>"#rgb"</code></strong>, <strong
class="userinput"><code>"#rrggbb"</code></strong> eller <strong
class="userinput"><code>"#rrrrggggbbbb"</code></strong>, där r, g och b är hexadecimala tal för de röda,
gröna och blåa komponenterna av färgen. Slutligen kan färgen också anges som en reell vektor som anger de
röda gröna och blåa komponenterna där komponenterna är mellan 0 och 1.</p><p>Fönstret ska som vanligt anges
som <strong
class="userinput"><code>[x1,x2,y1,y2,z1,z2]</code></strong>, eller kan alternativt anges som en sträng
<strong class="userinput"><code>"fit"</code></strong> i vilket fall x-intervallet kommer ställas in precis
och y-intervallet med fem procents gränser kring linjen.</p><p>Slutligen ska förklaring vara en sträng som
kan användas som förklaring i grafen. Det vill säga om förklaringar skrivs ut.</p><p>Exempel: </p><pre
class="screen"><code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)</code></strong>
<code class="prompt">genius></code> <strong
class="userinput"><code>SurfacePlotDrawPoints([0,0,0;1,-1,2;-1,-1,1])</code></strong>
</pre><p>Tillgängligt i version 1.0.19 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-VectorfieldClearSolutions"></a>VectorfieldClearSolutions</span></dt><dd><pre
class="synopsis">VectorfieldClearSolutions ()</pre><p>Rensar bort lösningarna som ritats av funktionen <a
class="link" href="ch11s20.html#gel-function-VectorfieldDrawSolution"><code
class="function">VectorfieldDrawSolution</code></a>.</p><p>Version 1.0.6 och framåt.</p></dd><dt><span
class="term"><a name="gel-function-VectorfieldDrawSolution"></a>VectorfieldDrawSolution</span></dt><dd><pre
class="synopsis">VectorfieldDrawSolution (x, y, dt, tlen)</pre><p>Då en verktorfältsgraf är aktiv, rita en
lösning med de angivna startvillkoren. Den vanliga Runge-Kutta-metoden används med ökning <code
class="varname">dt</code> under ett intervall med längden <code class="varname">tlen</code>. Lösningarna
stannar i grafen tills en annan graf visas eller tills du anropar <a class="link" href="ch11s20
.html#gel-function-VectorfieldClearSolutions"><code class="function">VectorfieldClearSolutions</code></a>.
Du kan också använda det grafiska gränssnittet för att rita lösningar och ange startvillkor med
musen.</p><p>Version 1.0.6 och framåt.</p></dd><dt><span class="term"><a
name="gel-function-VectorfieldPlot"></a>VectorfieldPlot</span></dt><dd><pre class="synopsis">VectorfieldPlot
(funkx, funky)</pre><pre class="synopsis">VectorfieldPlot (funkx, funky, x1, x2, y1, y2)</pre><p>Rita ett
tvådimensionellt vektorfält. Funktionen <code class="varname">funkx</code> ska vara dx/dt för vektorfältet
och funktionen <code class="varname">funky</code> ska vara dy/dt för vektorfältet. Funktionerna ska ta två
reella tal <code class="varname">x</code> och <code class="varname">y</code>, eller ett ensamt komplext tal.
Då parametern <a class="link" href="ch11s03.html#gel-function-VectorfieldNormalized"><code
class="function">VectorfieldNormalized</code></a> är <code class="con
stant">true</code> normaliseras magnituden för vektorerna. Det vill säga endast riktningen visas, inte
magnituden.</p><p>Valfritt kan du ange gränserna för graffönstret som <code class="varname">x1</code>, <code
class="varname">x2</code>, <code class="varname">y1</code>, <code class="varname">y2</code>. Om gränser inte
anges kommer de aktuellt inställda gränserna att användas (Se <a class="link"
href="ch11s03.html#gel-function-LinePlotWindow"><code
class="function">LinePlotWindow</code></a>).</p><p>Parametern <a class="link"
href="ch11s03.html#gel-function-LinePlotDrawLegends"><code class="function">LinePlotDrawLegends</code></a>
styr om förklaringen ritas ut.</p><p>Exempel: </p><pre class="screen"><code class="prompt">genius></code>
<strong class="userinput"><code>VectorfieldPlot(`(x,y)=x^2-y, `(x,y)=y^2-x, -1, 1, -1, 1)</code></strong>
</pre></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation
footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s19.html">Föregående</a> </td><td
width="20%" align="center"><a accesskey="u" href="ch11.html">Upp</a></td><td width="40%" align="right"> <a
accesskey="n" href="ch12.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top">Symboliska
operationer </td><td width="20%" align="center"><a accesskey="h" href="index.html">Hem</a></td><td
width="40%" align="right" valign="top"> Kapitel 12. Exempelprogram i GEL</td></tr></table></div></body></html>
diff --git a/help/sv/html/index.html b/help/sv/html/index.html
index d8f50cf..a1e3689 100644
--- a/help/sv/html/index.html
+++ b/help/sv/html/index.html
@@ -1,3 +1,62 @@
-<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Handbok för
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
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accesskey="n" href="ch01.html">Nästa</a></td></tr></table><hr></div><div lang="sv" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Handbok för Genius</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><span cla
ss="firstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Oklahoma State University<br></span><div class="address"><p> <code class="email"><<a
class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code> </p></div></div></div><div
class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">University of Queensland,
Australien<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:kaiw itee
uq edu au">kaiw itee uq edu au</a>></code> </p></div></div></div></div></div><div><p
class="releaseinfo">Denna handbok beskriver version 1.0.22 av Genius.</p></div><div><p
class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p class="copyright">Copyright ©
2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2016 Anders Jonsson (anders.jonsson@nors
jovallen.se)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>Tillstånd att kopiera,
distribuera och/eller modifiera detta dokument ges under villkoren i GNU Free Documentation License (GFDL),
version 1.1 eller senare, utgivet av Free Software Foundation utan standardavsnitt och omslagstexter. En
kopia av GFDL finns att hämta på denna <a class="ulink" href="ghelp:fdl" target="_top">länk</a> eller i filen
COPYING-DOCS som medföljer denna handbok.</p><p>Denna handbok utgör en av flera GNOME-handböcker som
distribueras under villkoren i GFDL. Om du vill distribuera denna handbok separat från övriga handböcker kan
du göra detta genom att lägga till en kopia av licensavtalet i handboken enligt instruktionerna i avsnitt 6 i
licensavtalet.</p><p>Många av namnen som används av företag för att särskilja deras produkter och tjänster är
registrerade varumärken. I de fall dessa namn förekommer i GNOME-dokumentation - och medlemmarna i GNOME-doku
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NÅGRA SOM HELST GARANTIER, VARE SIG UTTRYCKLIGA ELLER UNDERFÖRSTÅDDA, INKLUSIVE, MEN INTE BEGRÄNSAT TILL,
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class="legalnotice"><a name="idm45527219338448"></a><p
class="legalnotice-title"><b>Återkoppling</b></p><p>För att rapportera ett fel eller komma med ett förslag
för programmet <span class="application">Genius matematikverktyg</span> eller denna handbok, besök <a
class="ulink" href="http://www.jirka.org/genius.html"; target="_top">webbsidan för Genius</a> eller skicka mig
ett e-postmeddelande på <code class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z
com</a>></code>.</p></div></div><div><div class="revhistory"><table style="border-style:solid;
width:100%;" summary="Revisionshistorik"><tr><th align="left" valign="top" cols
pan="2"><b>Revisionshistorik</b></th></tr><tr><td align="left">Revision 0.2</td><td align="left">September
2016</td></tr><tr><td align="left" colspan="2">
+<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Handbok för
Genius</title><meta name="generator" content="DocBook XSL Stylesheets V1.79.1"><meta name="description"
content="Handbok för Genius matteverktyg."><link rel="home" href="index.html" title="Handbok för
Genius"><link rel="next" href="ch01.html" title="Kapitel 1. Introduktion"></head><body bgcolor="white"
text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%"
summary="Navigation header"><tr><th colspan="3" align="center">Handbok för Genius</th></tr><tr><td
width="20%" align="left"> </td><th width="60%" align="center"> </th><td width="20%" align="right"> <a
accesskey="n" href="ch01.html">Nästa</a></td></tr></table><hr></div><div lang="sv" class="book"><div
class="titlepage"><div><div><h1 class="title"><a name="index"></a>Handbok för Genius</h1></div><div><div
class="authorgroup"><div class="author"><h3 class="author"><span cla
ss="firstname">Jiří</span> <span class="surname">Lebl</span></h3><div class="affiliation"><span
class="orgname">Oklahoma State University<br></span><div class="address"><p> <code class="email"><<a
class="email" href="mailto:jirka 5z com">jirka 5z com</a>></code> </p></div></div></div><div
class="author"><h3 class="author"><span class="firstname">Kai</span> <span
class="surname">Willadsen</span></h3><div class="affiliation"><span class="orgname">University of Queensland,
Australien<br></span><div class="address"><p> <code class="email"><<a class="email" href="mailto:kaiw itee
uq edu au">kaiw itee uq edu au</a>></code> </p></div></div></div></div></div><div><p
class="releaseinfo">Denna handbok beskriver version 1.0.22 av Genius.</p></div><div><p
class="copyright">Copyright © 1997-2016 Jiří (George) Lebl</p></div><div><p class="copyright">Copyright ©
2004 Kai Willadsen</p></div><div><p class="copyright">Copyright © 2016 Anders Jonsson (anders.jonsson@nors
jovallen.se)</p></div><div><div class="legalnotice"><a name="legalnotice"></a><p>
+ Permission is granted to copy, distribute and/or modify this
+ document under the terms of the GNU Free Documentation
+ License (GFDL), Version 1.1 or any later version published
+ by the Free Software Foundation with no Invariant Sections,
+ no Front-Cover Texts, and no Back-Cover Texts. You can find
+ a copy of the GFDL at this <a class="ulink" href="ghelp:fdl" target="_top">link</a> or in the file
COPYING-DOCS
+ distributed with this manual.
+ </p><p> This manual is part of a collection of GNOME manuals
+ distributed under the GFDL. If you want to distribute this
+ manual separately from the collection, you can do so by
+ adding a copy of the license to the manual, as described in
+ section 6 of the license.
+ </p><p>
+ Many of the names used by companies to distinguish their
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+ names appear in any GNOME documentation, and the members of
+ the GNOME Documentation Project are made aware of those
+ trademarks, then the names are in capital letters or initial
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+ </p><p>
+ DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED
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+ </p></div></div><div><div class="legalnotice"><a name="idm51"></a><p
class="legalnotice-title"><b>Återkoppling</b></p><p>För att rapportera ett fel eller komma med ett förslag
för programmet <span class="application">Genius matematikverktyg</span> eller denna handbok, besök <a
class="ulink" href="http://www.jirka.org/genius.html"; target="_top">webbsidan för Genius</a> eller skicka mig
ett e-postmeddelande på <code class="email"><<a class="email" href="mailto:jirka 5z com">jirka 5z
com</a>></code>.</p></div></div><div><div class="revhistory"><table style="border-style:solid;
width:100%;" summary="Revisionshistorik"><tr><th align="left" valign="top"
colspan="2"><b>Revisionshistorik</b></th></tr><tr><td align="left">Revision 0.2</td><td
align="left">September 2016</td></tr><tr><td align="left" colspan="2">
<p class="author">Jiri (George) Lebl <code class="email"><<a class="email"
href="mailto:jirka 5z com">jirka 5z com</a>></code></p>
</td></tr></table></div></div><div><div class="abstract"><p
class="title"><b>Sammanfattning</b></p><p>Handbok för Genius
matteverktyg.</p></div></div></div><hr></div><div class="toc"><p><b>Innehållsförteckning</b></p><dl
class="toc"><dt><span class="chapter"><a href="ch01.html">1. Introduktion</a></span></dt><dt><span
class="chapter"><a href="ch02.html">2. Komma igång</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch02.html#genius-to-start">För att starta <span class="application">Genius
matematikverktyg</span></a></span></dt><dt><span class="sect1"><a href="ch02s02.html">Då du startar
Genius</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch03.html">3. Grundläggande
användning</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch03.html#genius-usage-workarea">Använda
arbetsytan</a></span></dt><dt><span class="sect1"><a href="ch03s02.html">För att skapa ett nytt
program</a></span></dt><dt><span class="sect1"><a href="ch03s03.html">Att ö
ppna eller köra ett program</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch04.html">4.
Grafritning</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch04.html#genius-line-plots">Linjegrafer</a></span></dt><dt><span class="sect1"><a
href="ch04s02.html">Parametriska grafer</a></span></dt><dt><span class="sect1"><a
href="ch04s03.html">Riktningsfältsgrafer</a></span></dt><dt><span class="sect1"><a
href="ch04s04.html">Vektorfältsgrafer</a></span></dt><dt><span class="sect1"><a
href="ch04s05.html">Ytgrafer</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch05.html">5.
Grunderna i GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch05.html#genius-gel-values">Värden</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-numbers">Tal</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-booleans">Booleska värden</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-
strings">Strängar</a></span></dt><dt><span class="sect2"><a
href="ch05.html#genius-gel-values-null">Null</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s02.html">Använda variabler</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s02.html#genius-gel-variables-setting">Ställa in variabler</a></span></dt><dt><span
class="sect2"><a href="ch05s02.html#genius-gel-variables-built-in">Inbyggda
variabler</a></span></dt><dt><span class="sect2"><a href="ch05s02.html#genius-gel-previous-result">Variabel
för föregående resultat</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch05s03.html">Använda
funktioner</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-defining">Definiera funktioner</a></span></dt><dt><span
class="sect2"><a href="ch05s03.html#genius-gel-functions-variable-argument-lists">Variabla
argumentlistor</a></span></dt><dt><span class="sect2"><a href="ch05s03.html#genius-gel-functions-passing-fu
nctions">Skicka funktioner till funktioner</a></span></dt><dt><span class="sect2"><a
href="ch05s03.html#genius-gel-functions-operations">Operationer på
funktioner</a></span></dt></dl></dd><dt><span class="sect1"><a
href="ch05s04.html">Avskiljare</a></span></dt><dt><span class="sect1"><a
href="ch05s05.html">Kommentarer</a></span></dt><dt><span class="sect1"><a
href="ch05s06.html">Moduloberäkning</a></span></dt><dt><span class="sect1"><a href="ch05s07.html">Lista över
GEL-operatorer</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch06.html">6. Programmering med
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch06.html#genius-gel-conditionals">Villkor</a></span></dt><dt><span class="sect1"><a
href="ch06s02.html">Slingor</a></span></dt><dd><dl><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-while">While-slingor</a></span></dt><dt><span class="sect2"><a
href="ch06s02.html#genius-gel-loops-for">For-slingor</a></span></dt><dt><span class="s
ect2"><a href="ch06s02.html#genius-gel-loops-foreach">Foreach-slingor</a></span></dt><dt><span
class="sect2"><a href="ch06s02.html#genius-gel-loops-break-continue">Break och
Continue</a></span></dt></dl></dd><dt><span class="sect1"><a href="ch06s03.html">Summor och
produkter</a></span></dt><dt><span class="sect1"><a
href="ch06s04.html">Jämförelseoperatorer</a></span></dt><dt><span class="sect1"><a
href="ch06s05.html">Globala variabler och räckvidd för variabler</a></span></dt><dt><span class="sect1"><a
href="ch06s06.html">Parametervariabler</a></span></dt><dt><span class="sect1"><a
href="ch06s07.html">Returnera</a></span></dt><dt><span class="sect1"><a
href="ch06s08.html">Referenser</a></span></dt><dt><span class="sect1"><a
href="ch06s09.html">Vvärden</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch07.html">7.
Avancerad programmering med GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch07.html#genius-gel-error-handling">Felhantering</a></span>
</dt><dt><span class="sect1"><a href="ch07s02.html">Toppnivåsyntax</a></span></dt><dt><span class="sect1"><a
href="ch07s03.html">Returnera funktioner</a></span></dt><dt><span class="sect1"><a
href="ch07s04.html">Verkligt lokala variabler</a></span></dt><dt><span class="sect1"><a
href="ch07s05.html">Uppstartsprocedur för GEL</a></span></dt><dt><span class="sect1"><a
href="ch07s06.html">Läsa in program</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch08.html">8.
Matriser i GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch08.html#genius-gel-matrix-support">Mata in matriser</a></span></dt><dt><span class="sect1"><a
href="ch08s02.html">Konjugattransponat och transponatoperator</a></span></dt><dt><span class="sect1"><a
href="ch08s03.html">Linjär algebra</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch09.html">9.
Polynom i GEL</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch09.html#genius-gel-polynomials-using">Använda polynom<
/a></span></dt></dl></dd><dt><span class="chapter"><a href="ch10.html">10. Mängdlära i
GEL</a></span></dt><dd><dl><dt><span class="sect1"><a href="ch10.html#genius-gel-sets-using">Använda
mängder</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch11.html">11. Lista över
GEL-funktioner</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch11.html#genius-gel-function-list-commands">Kommandon</a></span></dt><dt><span class="sect1"><a
href="ch11s02.html">Grundläggande</a></span></dt><dt><span class="sect1"><a
href="ch11s03.html">Parametrar</a></span></dt><dt><span class="sect1"><a
href="ch11s04.html">Konstanter</a></span></dt><dt><span class="sect1"><a href="ch11s05.html">Numeriska
funktioner</a></span></dt><dt><span class="sect1"><a
href="ch11s06.html">Trigonometri</a></span></dt><dt><span class="sect1"><a
href="ch11s07.html">Talteori</a></span></dt><dt><span class="sect1"><a
href="ch11s08.html">Matrismanipulation</a></span></dt><dt><span class="sect1"><a h
ref="ch11s09.html">Linjär algebra</a></span></dt><dt><span class="sect1"><a
href="ch11s10.html">Kombinatorik</a></span></dt><dt><span class="sect1"><a
href="ch11s11.html">Kalkyl</a></span></dt><dt><span class="sect1"><a
href="ch11s12.html">Funktioner</a></span></dt><dt><span class="sect1"><a
href="ch11s13.html">Ekvationslösning</a></span></dt><dt><span class="sect1"><a
href="ch11s14.html">Statistik</a></span></dt><dt><span class="sect1"><a
href="ch11s15.html">Polynom</a></span></dt><dt><span class="sect1"><a
href="ch11s16.html">Mängdlära</a></span></dt><dt><span class="sect1"><a href="ch11s17.html">Kommutativ
algebra</a></span></dt><dt><span class="sect1"><a href="ch11s18.html">Diverse</a></span></dt><dt><span
class="sect1"><a href="ch11s19.html">Symboliska operationer</a></span></dt><dt><span class="sect1"><a
href="ch11s20.html">Grafritning</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch12.html">12.
Exempelprogram i GEL</a></span></dt><dt><span class="cha
pter"><a href="ch13.html">13. Inställningar</a></span></dt><dd><dl><dt><span class="sect1"><a
href="ch13.html#genius-prefs-output">Utdata</a></span></dt><dt><span class="sect1"><a
href="ch13s02.html">Precision</a></span></dt><dt><span class="sect1"><a
href="ch13s03.html">Terminal</a></span></dt><dt><span class="sect1"><a
href="ch13s04.html">Minne</a></span></dt></dl></dd><dt><span class="chapter"><a href="ch14.html">14. Om <span
class="application">Genius matematikverktyg</span></a></span></dt></dl></div><div
class="list-of-figures"><p><b>Figurförteckning</b></p><dl><dt>2.1. <a
href="ch02s02.html#mainwindow-fig"><span class="application">Genius
matematikverktyg</span>-fönstret</a></dt><dt>4.1. <a href="ch04.html#lineplot-fig">Skapa
graf-fönster</a></dt><dt>4.2. <a href="ch04.html#lineplot2-fig">Graffönster</a></dt><dt>4.3. <a
href="ch04s02.html#paramplot-fig">Flik för parametriska grafer</a></dt><dt>4.4. <a
href="ch04s02.html#paramplot2-fig">Parametrisk graf</a></dt><d
t>4.5. <a href="ch04s05.html#surfaceplot-fig">Ytgraf</a></dt></dl></div></div><div
class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left">
</td><td width="20%" align="center"> </td><td width="40%" align="right"> <a accesskey="n"
href="ch01.html">Nästa</a></td></tr><tr><td width="40%" align="left" valign="top"> </td><td width="20%"
align="center"> </td><td width="40%" align="right" valign="top"> Kapitel 1.
Introduktion</td></tr></table></div></body></html>
diff --git a/help/update-xml-to-txt-html.sh b/help/update-xml-to-txt-html.sh
index abf1826..f9d7d90 100755
--- a/help/update-xml-to-txt-html.sh
+++ b/help/update-xml-to-txt-html.sh
@@ -12,6 +12,14 @@ LANGS="cs de el es fr pt_BR ru sv"
for n in $LANGS ; do
echo Running xml2po -e -p $n/$n.po -o $n/genius.xml C/genius.xml
xml2po -e -p $n/$n.po -o $n/genius.xml C/genius.xml || exit 1
+ echo Running rm -f $n/html/*.html
+ rm -f $n/html/*.html || exit 1
echo Running xmlto -o $n/html/ html $n/genius.xml
xmlto -o $n/html/ html $n/genius.xml || exit 1
done
+
+echo Running make-makefile-am.sh
+./make-makefile-am.sh
+
+echo
+echo Now you should rerun automake I suppose ...
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